# questions

tags: ['Physics', 'Current Electricity', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
title:
in-the-circuit-shown-in-figure-potential-difference-between-points-a-and-b-is-16-v-the-current-passing-through-2-resistance-will-be

slug:
in-the-circuit-shown-in-figure-potential-difference-between-points-a-and-b-is-16-v-the-current-passing-through-2-resistance-will-be-26758

question:
<div class="question">In the circuit shown in figure potential difference between points $A$ and $B$ is $16 \mathrm{~V}$. the current passing through $2 \Omega$ resistance will be
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/UZbUESIx80GM6K0l50vBIuvRhaCpq4e_7uhL3_sImFs.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.5 \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$3.5 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4.0 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">zero</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3.5 \mathrm{~A}$</span> </div>

answer:
<div class="solution">$\therefore 4 \mathrm{i}_{1}+2\left(\mathrm{i}_{1}+\mathrm{i}_{2}\right)-3+4 \mathrm{i}_{1}=16 \mathrm{~V}$
<br/> <br/>Using Kirchhoff's second law in the closed loop we have
<br/> <br/>$$
<br/> <br/>9-i_{2}-2\left(i_{1}+i_{2}\right)=0
<br/> <br/>$$
<br/> <br/>Solving equations (i) and (ii), we get $\mathrm{i}_{1}=1.5 \mathrm{~A}$ and $\mathrm{i}_{2}=2 \mathrm{~A}$
<br/> <br/>$\therefore$ current through $2 \mathrm{~W}$ resistor $=2+1.5=3.5 \mathrm{~A}$</div>


title: in-the-circuit-shown-in-the-figure-find-the-current-in-45-1
slug: in-the-circuit-shown-in-the-figure-find-the-current-in-45-1-86952
tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the circuit shown in the figure, find the current in $45 \Omega$.
<br/> <br/><img src=" data:image/png;base64,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"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.5 \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \mathrm{~A}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/k1gYpbYmFx4xl4y5OPK-kPU12RNgkYv570kmu78bSHI.original.fullsize.png"/><br/> <br/> <br/>$i=\frac{180}{90}=2 A$</div>


tags: ['Physics', 'Alternating Current', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
in-the-circuit-shown-in-the-figure-the-ac-source-gives-a-voltage-v-20-cos-2000-t-neglecting-source-resistance-the-voltmeter-and-ammeter-reading-will-b-1

slug:
in-the-circuit-shown-in-the-figure-the-ac-source-gives-a-voltage-v-20-cos-2000-t-neglecting-source-resistance-the-voltmeter-and-ammeter-reading-will-b-1-20854

question:
<div class="question">In the circuit shown in the figure, the $\mathrm{AC}$ source gives a voltage $V=20 \cos (2000 t)$ neglecting source resistance, the voltmeter and ammeter reading will be
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/v5y3bhAJ29Rq6lxNBnA5ho52xioZBvUCYER9Mb7lpWI.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$0 \mathrm{~V}, 0.47 \mathrm{~A}$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$282 \mathrm{~V}, 1.41 \mathrm{~A}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$1.41 \mathrm{~V}, 0.47 \mathrm{~A}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$15 \mathrm{~V}, 837 \mathrm{~A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$282 \mathrm{~V}, 1.41 \mathrm{~A}$
<br/></span> </div>

answer:
<div class="solution">Given, $R_1=8 \Omega$
<br/>$\begin{aligned} R_2 &amp; =2 \Omega \\ L &amp; =5 \mathrm{mH} \\ C &amp; =50 \mu \mathrm{F}\end{aligned}$
<br/>and $\quad V_0=20 \cos (2000 t)$
<br/>Impedance, $Z=\sqrt{R^2+\left(X_L-X_C\right)^2}$
<br/>As, here, $X_L=\omega L=2000 \times 5 \times 10^{-3}=10 \Omega$
<br/>Similarly, $X_L=\frac{1}{\omega C}=\frac{1}{2000 \times 50 \times 10^{-6}}=10 \Omega$ $\because \quad X_L=X_C$, Hence, $i_{\max }=\frac{V_0}{Z}=\frac{20}{(8+2)}=2 \mathrm{~A}$
<br/>Hence, $\quad i_{\operatorname{rms}}=\frac{i_{\max }}{\sqrt{2}}=\frac{2}{\sqrt{2}}=1.41 \mathrm{~A}$ and
<br/>$$
<br/>\begin{aligned}
<br/>V &amp; =R_2 i_{\mathrm{rms}}=1.41 \times 2 \\
<br/>&amp; =282 \mathrm{~V}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Alternating Current', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
in-the-circuit-shown-in-the-figure-the-ac-source-gives-a-voltage-v-20-cos-2000-t-neglecting-source-resistance-the-voltmeter-and-ammeter-reading-will-b

slug:
in-the-circuit-shown-in-the-figure-the-ac-source-gives-a-voltage-v-20-cos-2000-t-neglecting-source-resistance-the-voltmeter-and-ammeter-reading-will-b-99619

question:
<div class="question">In the circuit shown in the figure, the $\mathrm{AC}$ source gives a voltage $V=20 \cos (2000 t)$ neglecting source resistance, the voltmeter and ammeter reading will be
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/v5y3bhAJ29Rq6lxNBnA5ho52xioZBvUCYER9Mb7lpWI.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$0 \mathrm{~V}, 0.47 \mathrm{~A}$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$282 \mathrm{~V}, 1.41 \mathrm{~A}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$1.41 \mathrm{~V}, 0.47 \mathrm{~A}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$15 \mathrm{~V}, 837 \mathrm{~A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$282 \mathrm{~V}, 1.41 \mathrm{~A}$
<br/></span> </div>

answer:
<div class="solution">Given, $R_1=8 \Omega$
<br/>$\begin{aligned} R_2 &amp; =2 \Omega \\ L &amp; =5 \mathrm{mH} \\ C &amp; =50 \mu \mathrm{F}\end{aligned}$
<br/>and $\quad V_0=20 \cos (2000 t)$
<br/>Impedance, $Z=\sqrt{R^2+\left(X_L-X_C\right)^2}$
<br/>As, here, $X_L=\omega L=2000 \times 5 \times 10^{-3}=10 \Omega$
<br/>Similarly, $X_L=\frac{1}{\omega C}=\frac{1}{2000 \times 50 \times 10^{-6}}=10 \Omega$ $\because \quad X_L=X_C$, Hence, $i_{\max }=\frac{V_0}{Z}=\frac{20}{(8+2)}=2 \mathrm{~A}$
<br/>Hence, $\quad i_{\operatorname{rms}}=\frac{i_{\max }}{\sqrt{2}}=\frac{2}{\sqrt{2}}=1.41 \mathrm{~A}$ and
<br/>$$
<br/>\begin{aligned}
<br/>V &amp; =R_2 i_{\mathrm{rms}}=1.41 \times 2 \\
<br/>&amp; =282 \mathrm{~V}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Current Electricity', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
title:
in-the-circuit-shown-the-heat-produced-in-5-resistor-is-10-cal-s-1-the-heat-produced-per-sec-in-4-resistor-will-be

slug:
in-the-circuit-shown-the-heat-produced-in-5-resistor-is-10-cal-s-1-the-heat-produced-per-sec-in-4-resistor-will-be-33400

question:
<div class="question"><p>In the circuit shown, the heat produced in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math> resistor is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo> </mo><mi>cal</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. The heat produced per sec in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math> resistor will be</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/c9294d1a-a231-4ae9-958c-92963ee59679-image.png" style="width: 380px; height: 146px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi>cal</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>cal</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi>cal</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi>cal</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>cal</mi></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><msup><mi>I</mi><mn>2</mn></msup><mi>R</mi><mi>t</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfrac><msub><mi>H</mi><mn>1</mn></msub><msub><mi>H</mi><mn>2</mn></msub></mfrac><mo>=</mo><msup><mfenced><mfrac><msub><mi>l</mi><mn>1</mn></msub><msub><mi>I</mi><mn>2</mn></msub></mfrac></mfenced><mn>2</mn></msup><mo>×</mo><mfenced><mfrac><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>2</mn></msub></mfrac></mfenced></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>10</mn><msub><mi>H</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfenced close="]" open="["><mfrac><mn>1</mn><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced></mfrac></mfenced><mo>×</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mo> </mo><mi>cal</mi></math></p></div>


title: in-the-compound-ch-2-i-ch-ii-c-ch-iii-the-most-acidic-hydrogen-atom-is
slug: in-the-compound-ch-2-i-ch-ii-c-ch-iii-the-most-acidic-hydrogen-atom-is-92604
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question"><p>In the compound, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><msub><mi>CH</mi><mn>2</mn></msub><mrow><mo>(</mo><mi mathvariant="normal">I</mi><mo>)</mo></mrow></munder><mo>=</mo><munder><mi>CH</mi><mrow><mo>(</mo><mi>II</mi><mo>)</mo></mrow></munder><mo>-</mo><mi mathvariant="normal">C</mi><munder><mrow><mo>≡</mo><mi>CH</mi></mrow><mrow><mo>(</mo><mi>III</mi><mo>)</mo></mrow></munder></math></p><p>The most acidic hydrogen atom is</p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Only <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Only <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>II</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Only <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>III</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">All are equally acidic</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Only <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>III</mi></math></span> </div>

answer:
<div class="solution"><p>The hydrogen attached with most electronegative carbon will be the most acidic.</p><p>The hydrogen, attached with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sp</mi></math>-hybrid <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math>-atom is most acidic because of highest electronegativity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sp</mi></math> hybrid carbon atom, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">H</mi></math> atom becomes relatively more acidic.</p></div>


tags: ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
in-the-context-of-the-hallheroult-process-for-the-extraction-of-al-which-of-the-following-statement-is-incorrect

slug:
in-the-context-of-the-hallheroult-process-for-the-extraction-of-al-which-of-the-following-statement-is-incorrect-59128

question:
<div class="question">In the context of the Hall-Heroult process for the extraction of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Al</mi></math>, which of the following statement is incorrect.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>CO</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CO</mi><mn>2</mn></msub></math> are produced in the process.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Al</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>3</mn></msub></math> is mixed with <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CaF</mi><mn>2</mn></msub></math> which lowers the melting point of the mixture and brings conductivity.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Al</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup></math> is reduced at the cathode to form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Al</mi></math>.</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Na</mi><mn>3</mn></msub><msub><mi>AlF</mi><mn>6</mn></msub></math> serves as the electrolyte.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Na</mi><mn>3</mn></msub><msub><mi>AlF</mi><mn>6</mn></msub></math> serves as the electrolyte.</span> </div>

answer:
<div class="solution"><p>(i) In Hall-Heroult process for extraction of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Al</mi></math>, carbon anode is oxidised to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>CO</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CO</mi><mn>2</mn></msub></math>.</p><p>(ii) When <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Al</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>3</mn></msub></math> is mixed with <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CaF</mi><mn>2</mn></msub></math>, it lowers the melting point of the mixture and brings conductivity.</p><p>(iii) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Al</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup></math> is reduced at cathode to form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Al</mi></math>.</p><p>(iv) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Al</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>3</mn></msub></math> serves as the electrolyte, undergoing the redox process. <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Na</mi><mn>3</mn></msub><msub><mi>AlF</mi><mn>6</mn></msub></math> although is an electrolyte but serves as a solvent, not electrolyte.</p></div>


tags: ['Physics', 'Laws of Motion', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
in-the-diagram-shown-below-both-the-strings-a-b-and-c-d-are-made-of-same-material-and-have-same-crosssection-the-pulleys-are-light-and-fictionless-if-

slug:
in-the-diagram-shown-below-both-the-strings-a-b-and-c-d-are-made-of-same-material-and-have-same-crosssection-the-pulleys-are-light-and-fictionless-if-25047

question:
<div class="question">In the diagram shown below, both the strings $A B$ and $C D$ are made of same material and have same cross-section. The pulleys are light and fictionless. If the speed of wave in string $A B$ is $v_1$ and in $C D$ is $v_2$, then $\frac{v_1}{v_2}$ is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/KcmsJruf_zJD-FmbOZ5Tbb2sh-_WbMzr1hs2wiLMZpU.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>1 
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$\sqrt{2}$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>2
<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"> <br/>$1 / \sqrt{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$1 / \sqrt{2}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/WLHC607-XBSHYYUfIPQtPzzvqoOMBp9UjPn556gClac.original.fullsize.png"/><br/> <br/>Let the speed of wave in string be $v$ and tension in string be $T$.
<br/>$$
<br/>\begin{gathered}
<br/>v \propto \sqrt{T} \\
<br/>\frac{v_{A B}}{v_{C D}}=\sqrt{\frac{T}{2 T}}=\frac{1}{\sqrt{2}}
<br/>\end{gathered}
<br/>$$</div>


title: in-the-equation-pv-constant-the-value-of-is-unity-then-the-process-is
slug: in-the-equation-pv-constant-the-value-of-is-unity-then-the-process-is-14218
tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2010']
answer: <div class="solution">$\mathrm{PV}=$ constant represents isothermal process.</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">In the equation $\mathrm{PV}^{\gamma}=$ constant, the value of $\gamma$ is unity. Then the process is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">isothermal</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">adiabatic</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">isobaric</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">irreversible</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">isothermal</span> </div>


title: in-the-expansion-of-1-x-x-3-x-4-10-the-coefficient-of-x-4-is
slug: in-the-expansion-of-1-x-x-3-x-4-10-the-coefficient-of-x-4-is-80737
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow></mfenced><mn>10</mn></msup></math>, the coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>235</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>310</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>285</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>325</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>310</mn></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow></mfenced><mn>10</mn></msup><mo>=</mo><msup><mfenced close="]" open="["><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></mrow></mfenced><mn>10</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>10</mn></msup><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mn>10</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>10</mn></mmultiscripts><mi>x</mi><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>10</mn></mmultiscripts><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mmultiscripts><mi>C</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>10</mn></mmultiscripts><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mmultiscripts><mi>C</mi><mn>4</mn><none></none><mprescripts></mprescripts><none></none><mn>10</mn></mmultiscripts><msup><mi>x</mi><mn>4</mn></msup><mo>…</mo></mrow></mfenced><mo>×</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>10</mn></mmultiscripts><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>10</mn></mmultiscripts><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mo>…</mo><mo>.</mo></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>=</mo><mfenced><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>10</mn></mmultiscripts></mfenced><mfenced><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>10</mn></mmultiscripts></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>4</mn><none></none><mprescripts></mprescripts><none></none><mn>10</mn></mmultiscripts></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>100</mn><mo>+</mo><mn>210</mn><mo>=</mo><mn>310</mn></math></p></div>


tags: ['Physics', 'Capacitance', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
title:
in-the-figure-below-what-is-the-potential-difference-between-the-point-a-and-b-and-between-b-and-c-respectively-in-steady-state

slug:
in-the-figure-below-what-is-the-potential-difference-between-the-point-a-and-b-and-between-b-and-c-respectively-in-steady-state-40853

question:
<div class="question">In the figure below, what is the potential difference between the point $A$ and $B$ and between $B$ and $C$ respectively in steady state
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/05n1XZBNDOJvFz3awJy97ZmZHCfchGzAyeAunMvSmJM.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad V_{A B}=V_{B C}=100 \mathrm{~V}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad V_{A B}=75 V, V_{B C}=25 V$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\quad V_{A B}=25 V, V_{B C}=75 V$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad V_{A B}=V_{B C}=50 \mathrm{~V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad V_{A B}=25 V, V_{B C}=75 V$</span> </div>

answer:
<div class="solution">The equivalent circuit is shown in figure.
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>V_{1}+V_{2}=100 \\
<br/> <br/>\text { and } 2 V_{1}=6 V_{2}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/s2O76azAVpnq35rd4-HqwdGjim3m6EWgcmopPQHMzZQ.original.fullsize.png"/><br/> <br/> <br/>On solving above equations, we get $V_{1}=75 V, V_{2}=25 V$</div>


title: in-the-following-circuit-diagram-the-current-through-the-battery-is
slug: in-the-following-circuit-diagram-the-current-through-the-battery-is-68265
tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question"><p>In the following circuit diagram, the current through the battery is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/fcbfcbd8-29e0-45c2-9d2c-d24c806846cf-image.png" style="width: 180px; height: 217px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </div>

answer:
<div class="solution"><p>From the figure, It is clear that diodes <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mn>3</mn></msub></math> are reverse biased, therefore they will not conduct any current and the resistance of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math> will be not in use. The diode <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mn>2</mn></msub></math> is forward biased, therefore the current flowing through the diode <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mn>2</mn></msub></math> is same as the current through the battery.</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/6de3e7c2-d986-4868-a896-ff5a78832d95-image.png" style="width: 270px; height: 180px;"/></p><p>On redrawing the circuit as shown, we can obtain the value of current as</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mfrac><mn>20</mn><mrow><mn>7</mn><mo>+</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mo> </mo><mi mathvariant="normal">A</mi></math></p></div>


title: in-the-following-circuit-the-equivalent-resistance-between-x-and-y-is
slug: in-the-following-circuit-the-equivalent-resistance-between-x-and-y-is-40061
tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the following circuit the equivalent resistance between $X$ and $Y$ is ...... $\Omega$ 
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/YcXXBDduEXuRqVDos-e62MtGTtBLVkpqTFOrTOWv5jc.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$5$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$12$
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$16$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$20$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$16$
<br/></span> </div>

answer:
<div class="solution">According to the given figure $X$ is at lower potential w.r.t $Y$. Hence, both diodes are in reverse biasing, so, equivalent circuit can be redrawn as follows
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/sLfOWDF4NSUgD7_vpFYr_vPdbZFOx-QJjBZkv2zJGKY.original.fullsize.png"/><br/> <br/>Equivalent resistance between $X$ and $Y$
<br/>$$
<br/>R_{\text {eq }}=8+2+6=16 \Omega
<br/>$$</div>


tags: ['Physics', 'Electrostatics', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
in-the-following-figures-which-figure-represents-the-variation-of-the-electric-field-with-distance-from-the-centre-of-a-uniformly-charged-nonconductin-1

slug:
in-the-following-figures-which-figure-represents-the-variation-of-the-electric-field-with-distance-from-the-centre-of-a-uniformly-charged-nonconductin-1-74325

question:
<div class="question">In the following figures, which figure represents the variation of the electric field with distance from the centre of a uniformly charged non-conducting spheres of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/4fb09fe8-09cd-4c4f-8c11-10e643f54d14-image.png" style="width: 220px; height: 188px;"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/1935f8e6-c845-49a1-8a8e-ebd12c7640e2-image.png" style="width: 220px; height: 187px;"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/2cd268d1-a051-4ff8-86e0-ca7e471e9e05-image.png" style="width: 220px; height: 188px;"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ca443cf8-3d12-4ee1-8f56-869e5a98be17-image.png" style="width: 220px; height: 188px;"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/2cd268d1-a051-4ff8-86e0-ca7e471e9e05-image.png" style="width: 220px; height: 188px;"/></span> </div>

answer:
<div class="solution"><p>The electric field intensity at a point lying outside the sphere (non-conducting) is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub></mrow></mfrac><mo>·</mo><mfrac><mi>q</mi><msup><mi>r</mi><mn>2</mn></msup></mfrac></math></p><p>i.e.  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>∝</mo><mfrac><mn>1</mn><msup><mi>r</mi><mn>2</mn></msup></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>…</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>The electric field intensity at the surface <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>r</mi><mo>=</mo><mi>R</mi><mo>)</mo></math>,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub></mrow></mfrac><mfrac><mi>q</mi><msup><mi>R</mi><mn>2</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>E</mi><mo>∝</mo><mfrac><mn>1</mn><msup><mi>R</mi><mn>2</mn></msup></mfrac></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> being the radius of sphere,</p><p>The electric field intensity inside the sphere is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mfrac><mrow><mi>q</mi><mi>r</mi></mrow><mrow><mn>4</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msup><mi>R</mi><mn>3</mn></msup></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>E</mi><mo>∝</mo><mi>r</mi></math></p><p>At the centre of sphere, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo>∴</mo><mi>E</mi><mo>=</mo><mn>0</mn></math></p></div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
in-the-following-figures-which-figure-represents-the-variation-of-the-electric-field-with-distance-from-the-centre-of-a-uniformly-charged-nonconductin

slug:
in-the-following-figures-which-figure-represents-the-variation-of-the-electric-field-with-distance-from-the-centre-of-a-uniformly-charged-nonconductin-19033

question:
<div class="question">In the following figures, which figure represents the variation of the electric field with distance from the centre of a uniformly charged non-conducting spheres of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/4fb09fe8-09cd-4c4f-8c11-10e643f54d14-image.png" style="width: 220px; height: 188px;"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/1935f8e6-c845-49a1-8a8e-ebd12c7640e2-image.png" style="width: 220px; height: 187px;"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/2cd268d1-a051-4ff8-86e0-ca7e471e9e05-image.png" style="width: 220px; height: 188px;"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ca443cf8-3d12-4ee1-8f56-869e5a98be17-image.png" style="width: 220px; height: 188px;"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/2cd268d1-a051-4ff8-86e0-ca7e471e9e05-image.png" style="width: 220px; height: 188px;"/></span> </div>

answer:
<div class="solution"><p>The electric field intensity at a point lying outside the sphere (non-conducting) is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub></mrow></mfrac><mo>·</mo><mfrac><mi>q</mi><msup><mi>r</mi><mn>2</mn></msup></mfrac></math></p><p>i.e.  <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>∝</mo><mfrac><mn>1</mn><msup><mi>r</mi><mn>2</mn></msup></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>…</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>The electric field intensity at the surface <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>r</mi><mo>=</mo><mi>R</mi><mo>)</mo></math>,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub></mrow></mfrac><mfrac><mi>q</mi><msup><mi>R</mi><mn>2</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>E</mi><mo>∝</mo><mfrac><mn>1</mn><msup><mi>R</mi><mn>2</mn></msup></mfrac></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> being the radius of sphere,</p><p>The electric field intensity inside the sphere is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mfrac><mrow><mi>q</mi><mi>r</mi></mrow><mrow><mn>4</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msup><mi>R</mi><mn>3</mn></msup></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>E</mi><mo>∝</mo><mi>r</mi></math></p><p>At the centre of sphere, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo>∴</mo><mi>E</mi><mo>=</mo><mn>0</mn></math></p></div>


tags: ['Physics', 'Capacitance', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
title:
in-the-given-circuit-diagram-e-5-v-r-1-r-2-4-r-1-r-3-1-and-c-3-f-then-what-will-be-the-numerical-value-of-charge-on-each-plates-of-the-capacitor

slug:
in-the-given-circuit-diagram-e-5-v-r-1-r-2-4-r-1-r-3-1-and-c-3-f-then-what-will-be-the-numerical-value-of-charge-on-each-plates-of-the-capacitor-82813

question:
<div class="question"><p>In the given circuit diagram,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">V</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mo> </mo><msub><mi>R</mi><mn>2</mn></msub><mo>=</mo><mn>4</mn><mo> </mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mo> </mo><msub><mi>R</mi><mn>1</mn></msub><mo>=</mo><msub><mi>R</mi><mn>3</mn></msub><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mn>3</mn><mo> </mo><mi>μ</mi><mi mathvariant="normal">F</mi></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/5ad9a203-a9be-4e03-bbae-704afa819346-image.png" style="width: 220px; height: 160px;"/></p><p>Then, what will be the numerical value of charge on each plates of the capacitor.</p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24</mn><mo> </mo><mi>μC</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo> </mo><mi>μC</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mi>μC</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi>μC</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mi>μC</mi></math></span> </div>

answer:
<div class="solution"><p>Current will flow only through the branch containing resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>2</mn></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>i</mi><mo>=</mo><mfrac><mi>E</mi><mrow><msub><mi>R</mi><mn>2</mn></msub><mo>+</mo><mi>r</mi></mrow></mfrac><mo>=</mo><mfrac><mn>5</mn><mrow><mn>4</mn><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">A</mi></math></p><p>Potential difference across <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>2</mn></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>4</mn><mo> </mo><mi mathvariant="normal">V</mi></math></p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> be the charge on each plate of capacitor, then</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>q</mi><mi>c</mi></mfrac><mo>+</mo><mfrac><mi>q</mi><mi>c</mi></mfrac><mo>=</mo><mn>4</mn></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>q</mi></mrow><mi>c</mi></mfrac><mo>=</mo><mn>4</mn></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>q</mi></mrow><mrow><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>6</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn>4</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>6</mn><mo> </mo><mi>μC</mi></math></p></div>


tags: ['Physics', 'Current Electricity', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
in-the-given-circuit-it-is-observed-that-the-current-i-is-independent-of-the-value-of-resistance-r-5-then-the-resistance-value-must-satisfy

slug:
in-the-given-circuit-it-is-observed-that-the-current-i-is-independent-of-the-value-of-resistance-r-5-then-the-resistance-value-must-satisfy-97183

question:
<div class="question"><p>In the given circuit, it is observed that the current <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math> is independent of the value of resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>5</mn></msub></math>. Then, the resistance value must satisfy</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/2d404985-8a83-4f44-9f6a-e17b22cb3318-image.png" style="width: 250px; height: 181px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msub><mi>R</mi><mn>5</mn></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>R</mi><mn>6</mn></msub></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>1</mn></msub><mo>+</mo><msub><mi>R</mi><mn>2</mn></msub></mrow></mfrac><mo>·</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>3</mn></msub><mo>+</mo><msub><mi>R</mi><mn>4</mn></msub></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>4</mn></msub><mo>=</mo><msub><mi>R</mi><mn>2</mn></msub><msub><mi>R</mi><mn>3</mn></msub></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>2</mn></msub><msub><mi>R</mi><mn>5</mn></msub><mo>=</mo><msub><mi>R</mi><mn>3</mn></msub><msub><mi>R</mi><mn>4</mn></msub><msub><mi>R</mi><mn>6</mn></msub></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>3</mn></msub><mo>=</mo><msub><mi>R</mi><mn>2</mn></msub><msub><mi>R</mi><mn>4</mn></msub><mo>=</mo><msub><mi>R</mi><mn>5</mn></msub><msub><mi>R</mi><mn>6</mn></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>4</mn></msub><mo>=</mo><msub><mi>R</mi><mn>2</mn></msub><msub><mi>R</mi><mn>3</mn></msub></math></span> </div>

answer:
<div class="solution"><p>From the given circuit, we can say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math> is independent of resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>5</mn></msub><mo>,</mo></math> so no current flows through <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>5</mn></msub><mo>.</mo></math> This required that the junction of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>2</mn></msub></math> is at the same potential as the junction of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>3</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>4</mn></msub></math> (Wheatstone's bridge balancing condition)</p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>R</mi><mn>3</mn></msub><msub><mi>R</mi><mn>4</mn></msub></mfrac></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>4</mn></msub><mo>=</mo><msub><mi>R</mi><mn>3</mn></msub><msub><mi>R</mi><mn>2</mn></msub></math></p></div>


title: in-the-given-circuit-rms-value-of-current-irms-through-the-resistor-r-is-1
slug: in-the-given-circuit-rms-value-of-current-irms-through-the-resistor-r-is-1-91867
tags: ['Physics', 'Alternating Current', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the given circuit, rms value of current (Irms) through the resistor $R$ is :<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/aHaZk2pei9iFEQ9YHqBkmjmbU2nf5eV9CESgR6GnNIM.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2} \mathrm{~A}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$20 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \sqrt{2} \mathrm{~A}$</span> </li> </ul>

answer:
<div class="solution">(a) Given,<br/>Capacitive reactance, $\mathrm{X}_{\mathrm{C}}=100 \Omega$<br/>Inductive reactance, $\mathrm{X}_{\mathrm{L}}=200 \Omega$<br/>Resistance, $R=100 \Omega$<br/>Impedance, $\mathrm{Z}=\sqrt{\mathrm{R}^2+\left(\mathrm{X}_{\mathrm{L}}-\mathrm{X}_{\mathrm{C}}\right)^2}$<br/>$Z=\sqrt{100^2+(200-100)^2}=100 \sqrt{2} \Omega$<br/>RMS value of current,<br/>$i_{\mathrm{rms}}=\frac{V_{\mathrm{rms}}}{Z}=\frac{200 \sqrt{2}}{100 \sqrt{2}}=2 \mathrm{~A}$</div>


title: in-the-given-circuit-the-current-through-battery-is
slug: in-the-given-circuit-the-current-through-battery-is-56108
tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
question:
<div class="question"><p>In the given circuit, The current through battery is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/99330477-c4f5-4504-8cc4-746954fc6bb8-image.png" style="width: 270px; height: 205px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </div>

answer:
<div class="solution"><p>In the given circuit, diode <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mn>1</mn></msub></math> is reverse biased, so it will not conduct but <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mn>2</mn></msub><mo>,</mo><msub><mi>D</mi><mn>3</mn></msub></math> are forward biased, so they will conduct, hence corresponding equivalent circuit. For the given circuit is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/f26c2ab3-ada7-462b-bf4b-ba19fef8300c-image.png" style="width: 300px; height: 227px;"/></p><p>Now, the equivalent resistance of the circuit is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>eq</mi></msub><mo>=</mo><mfrac><mrow><mfenced><mn>515</mn></mfenced><mo>×</mo><mn>20</mn></mrow><mrow><mfenced><mrow><mn>5</mn><mo>+</mo><mn>5</mn></mrow></mfenced><mo>+</mo><mn>20</mn></mrow></mfrac><mo>+</mo><mfrac><mn>20</mn><mn>3</mn></mfrac><mo> </mo><mi mathvariant="normal">Ω</mi></math></p><p>Current through battery, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mfrac><mrow><mn>10</mn><mo> </mo><mi mathvariant="normal">V</mi></mrow><mrow><mfrac><mn>20</mn><mn>3</mn></mfrac><mo> </mo><mi mathvariant="normal">Ω</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">A</mi></math></p></div>


tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
title:
in-the-given-figure-the-equation-of-the-larger-circle-is-x-2-y-2-4-y-5-0-and-the-distance-between-centres-is-4-then-the-equation-of-smaller-circle-is

slug:
in-the-given-figure-the-equation-of-the-larger-circle-is-x-2-y-2-4-y-5-0-and-the-distance-between-centres-is-4-then-the-equation-of-smaller-circle-is-14862

question:
<div class="question">In the given figure, the equation of the larger circle is $x^{2}+y^{2}+4 y-5=0$ and the distance between centres is 4 . Then the equation of smaller circle is
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/XO5x82sLhpvVZjtM7P65tDrKkKEzFvjh-JRXhpSCYKw.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(x-\sqrt{7})^{2}+(y-1)^{2}=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad(x+\sqrt{7})^{2}+(y-1)^{2}=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}+y^{2}=2 \sqrt{7} x+2 y$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(x-\sqrt{7})^{2}+(y-1)^{2}=1$</span> </div>

answer:
<div class="solution">We have $x^{2}+y^{2}+4 y-5=0$. Its centre is $\mathrm{C}_{1}(0,-2)$
<br/> <br/>$\mathrm{r}_{1}=\sqrt{4+5}=3$. Let $\mathrm{C}_{2}(\mathrm{~h}, \mathrm{k})$ be the centre of the
<br/> <br/>smaller circle and its radius $\mathrm{r}_{2}$. Then $\mathrm{C}_{1} \mathrm{C}_{2}=4$ $\Rightarrow \sqrt{\mathrm{h}^{2}+(\mathrm{k}+2)^{2}}=3+\mathrm{r}_{2}=4$
<br/> <br/>$\Rightarrow \mathrm{r}_{2}=1$
<br/> <br/>But $\mathrm{k}=\mathrm{r}_{2}=1 \quad$ [it touches $\mathrm{x}$ -axis
<br/> <br/>$\therefore$ From eq $(1), 4=\sqrt{\mathrm{h}^{2}+(1+2)^{2}}$
<br/> <br/>$\Rightarrow 16=\mathrm{h}^{2}+9 \Rightarrow \mathrm{h}^{2}=7 \Rightarrow \mathrm{h}=\pm \sqrt{7}$
<br/> <br/>Since $\mathrm{h}&gt;0 \quad \therefore \mathrm{h}=\sqrt{7}$
<br/> <br/>Hence, required circle is $(x-\sqrt{7})^{2}+(y-1)^{2}=1$</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'BITSAT', 'BITSAT 2014']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P1.db
title:
in-the-kinetic-theory-of-gases-which-of-these-statements-isare-true-i-the-pressure-of-a-gas-is-proportional-to-the-mean-speed-of-the-molecules-ii-the-

slug:
in-the-kinetic-theory-of-gases-which-of-these-statements-isare-true-i-the-pressure-of-a-gas-is-proportional-to-the-mean-speed-of-the-molecules-ii-the-23531

question:
<div class="question">In the kinetic theory of gases, which of these statements is/are true?
<br/> <br/>(i) The pressure of a gas is proportional to the mean speed of the molecules.
<br/> <br/>(ii) The root mean square speed of the molecules is proportional to the pressure.
<br/> <br/>(iii) The rate of diffusion is proportional to the mean speed of the molecules.
<br/> <br/>(iv) The mean translational kinetic energy of a gas is proportional to its kelvin temperature.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">(ii) and (iii) only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">(i),(ii) and (iv) only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">(i) and (iii) only</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">(iii) and (iv) only</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">(iii) and (iv) only</span> </div>


title: in-the-laboratory-manganese-ii-salt-is-oxidise-to-permanganate-ion-in-aqueous-solution-by
slug: in-the-laboratory-manganese-ii-salt-is-oxidise-to-permanganate-ion-in-aqueous-solution-by-76389
tags: ['Chemistry', 'd and f Block Elements', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the laboratory, manganese (II) salt is oxidise to permanganate ion in aqueous solution by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">hydrogen peroxide</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">conc. nitric acid</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">peroxy disulphate</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">dichromate</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">peroxy disulphate</span> </div>

answer:
<div class="solution">In laboratory, manganese (II) ion salt is oxidised to permanganate ion in aqueous solution by peroxodisulphate.<br/>$2 \mathrm{Mn}^{2+}+\underset{\text{peroxodisulphate ion}}{5 \mathrm{~S}_2 \mathrm{O}_8^{2-}}+8 \mathrm{H}_2 \mathrm{O} \rightarrow$ $2 \mathrm{MnO}_4^{-}+10 \mathrm{SO}_4^{2-}+16 \mathrm{H}^{+}$</div>


title: in-the-manufacture-of-iron-from-haematite-limestone-is-added-to-act-as
slug: in-the-manufacture-of-iron-from-haematite-limestone-is-added-to-act-as-58672
tags: ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Flux</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">In the manufacture of iron from haematite, limestone is added to act as :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Flux</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Slag</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">A reducing agent</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">An oxidizing agent</span> </li> </ul>

answer:
<div class="solution">Limestone $\left(\mathrm{CaCO}_{3}\right)$ is mixed with $\mathrm{Fe}_{2} \mathrm{O}_{3}$ and it acts as flux to form slag $\left(\mathrm{CaSiO}_{3}\right)$.</div>


title: in-the-molecules-ch-4-nf-3-nh-4-and-h-2-o
slug: in-the-molecules-ch-4-nf-3-nh-4-and-h-2-o-99221
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the molecules $\mathrm{CH}_4, \mathrm{NF}_3, \mathrm{NH}_4^{+}$and $\mathrm{H}_2 \mathrm{O}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>number of lone pairs are same
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>all have same hybridisation of centre of atom 
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>the bond angles are same 
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>number of bond pairs are same</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>all have same hybridisation of centre of atom 
<br/></span> </div>

answer:
<div class="solution">$\mathrm{CH}_4 ; \mathrm{bp}=4 ; s p^3$-hybridisation
<br/>$\mathrm{NF}_3 ; \mathrm{bp}=3 ; l p=1 ; s p^3$-hybridisation
<br/>$\mathrm{NH}_4^{+} ; \mathrm{bp}=4 ; l p=0 ; s p^3$-hybridisation
<br/>$\mathrm{H}_2 \mathrm{O} ; \mathrm{bp}=2 ; l p=2 ; s p^3$-hybridisation
<br/>Hence, all have the same slope.</div>


title: in-the-periodic-table-with-the-increase-in-atomic-number-the-metallic-character-of-an-element
slug: in-the-periodic-table-with-the-increase-in-atomic-number-the-metallic-character-of-an-element-49502
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'BITSAT', 'BITSAT 2010']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">In the periodic table, with the increase in atomic number, the metallic character of an element</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">decreases in a period and increases in a group</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">increases in a period and decreases in a group</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">increases both in a period and the group</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">decreases in a period and the group</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">decreases in a period and increases in a group</span> </div>


title: in-the-reaction-2-pcl-5-pcl-4-pcl-6-the-change-in-hybridisation-is-from
slug: in-the-reaction-2-pcl-5-pcl-4-pcl-6-the-change-in-hybridisation-is-from-17506
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the reaction $2 \mathrm{PCl}_{5} \rightleftharpoons \mathrm{PCl}_{4}^{+}+\mathrm{PCl}_{6}^{-}$, the change in hybridisation is from</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$s p^{3} d$ to $s p^{3}$ and $s p^{3} d^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$s p^{3} d$ to $s p^{2}$ and $s p^{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$s p^{3} d$ to $s p^{3} d^{2}$ and $s p^{3} d^{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$s p^{3} d^{2}$ to $s p^{3}$ and $s p^{3} d$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$s p^{3} d$ to $s p^{3}$ and $s p^{3} d^{2}$</span> </div>

answer:
<div class="solution">$2 \mathrm{PCl}_{5} \rightleftharpoons \mathrm{PCl}_{4}^{+}+\mathrm{PCl}_{6}^{-}$
<br/> <br/>$\uparrow s p^{3} d \quad \uparrow s p^{3} \quad \uparrow s p^{3} d^{2}$</div>


title: in-the-reaction-3-br-2-6-co-3-2-3-h-2-o-5-br-bro-3-6-hco-3
slug: in-the-reaction-3-br-2-6-co-3-2-3-h-2-o-5-br-bro-3-6-hco-3-75102
tags: ['Chemistry', 'Redox Reactions', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the reaction
<br/>$3 \mathrm{Br}_{2}+6 \mathrm{CO}_{3}^{2-}+3 \mathrm{H}_{2} \mathrm{O} \rightarrow 5 \mathrm{Br}^{-}+\mathrm{BrO}_{3}^{-}+6 \mathrm{HCO}_{3}^{-}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Bromine is oxidised and carbonate is reduced.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Bromine is reduced and water is oxidised.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Bromine is neither reduced nor oxidised.</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Bromine is both reduced and oxidised.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Bromine is both reduced and oxidised.</span> </div>

answer:
<div class="solution">$\begin{aligned}
<br/>3 \mathrm{Br}_{2}+6 \mathrm{CO}_{3}^{2-}+3 \mathrm{H}_{2} \mathrm{O} &amp; \rightarrow \\
<br/>5 \mathrm{Br}^{-} &amp;+\mathrm{BrO}_{3}^{-}+6 \mathrm{HCO}_{3}^{-}
<br/>\end{aligned}$
<br/>O.N. of $\mathrm{Br}_{2}$ changes from 0 to -1 and +5 , hence it is reduced as well as oxidised.</div>


title: in-the-reaction-ch-3-ch-2-cl-kcn-a-lialh-4-p-end-product-p-is
slug: in-the-reaction-ch-3-ch-2-cl-kcn-a-lialh-4-p-end-product-p-is-30699
tags: ['Chemistry', 'Amines', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question"><p>In the reaction, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><mo>·</mo><msub><mi>CH</mi><mn>2</mn></msub><mo>·</mo><mi>Cl</mi><mover><mo>⟶</mo><mi>KCN</mi></mover><mfenced><mi mathvariant="normal">A</mi></mfenced><mover><mo>⟶</mo><msub><mi>LiAlH</mi><mn>4</mn></msub></mover><mfenced><mi mathvariant="normal">P</mi></mfenced></math> end product <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">P</mi></mfenced></math> is</p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><msub><mi>CH</mi><mn>2</mn></msub><mo>·</mo><msub><mi>NO</mi><mn>2</mn></msub></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><msub><mi>CH</mi><mn>2</mn></msub><msub><mi>CH</mi><mn>2</mn></msub><msub><mi>NO</mi><mn>2</mn></msub></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><msub><mi>CH</mi><mn>2</mn></msub><msub><mi>NH</mi><mn>2</mn></msub></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><mo>·</mo><msub><mi>CH</mi><mn>2</mn></msub><mo>·</mo><msub><mi>CH</mi><mn>2</mn></msub><msub><mi>NH</mi><mn>2</mn></msub></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><mo>·</mo><msub><mi>CH</mi><mn>2</mn></msub><mo>·</mo><msub><mi>CH</mi><mn>2</mn></msub><msub><mi>NH</mi><mn>2</mn></msub></math></span> </div>

answer:
<div class="solution"><p>When the reducing agent is lithium aluminium hydride, the aldehyde group is reduced to primary alcohol but the carbon-carbon double bond is unaffected.</p><p>In the given reaction the end product is (d). It is obtained<br/>as follows.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><msub><mi>CH</mi><mn>2</mn></msub><mo>·</mo><mi>Cl</mi><mover><mo>⟶</mo><mi>KCN</mi></mover><munder><mrow><msub><mi>CH</mi><mn>3</mn></msub><msub><mi>CH</mi><mn>2</mn></msub><mi>CN</mi></mrow><mrow><mo>(</mo><mi mathvariant="normal">A</mi><mo>)</mo></mrow></munder><munderover><mo>→</mo><mrow><mo>(</mo><mi>Reduction</mi><mo>)</mo></mrow><msub><mi>LiAlH</mi><mn>4</mn></msub></munderover><msub><mi>CH</mi><mn>3</mn></msub><msub><mi>CH</mi><mn>2</mn></msub><msub><mi>CH</mi><mn>2</mn></msub><mo>·</mo><msub><mi>NH</mi><mn>2</mn></msub></math></p></div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
title:
in-the-reaction-of-kmno-4-with-an-oxalate-in-acidic-medium-mno-4-is-reduced-to-mn-2-and-c-2-o-4-2-is-oxidised-to-co-2-hence-50-ml-of-002-m-kmno-4-is-e

slug:
in-the-reaction-of-kmno-4-with-an-oxalate-in-acidic-medium-mno-4-is-reduced-to-mn-2-and-c-2-o-4-2-is-oxidised-to-co-2-hence-50-ml-of-002-m-kmno-4-is-e-31423

question:
<div class="question">In the reaction of $\mathrm{KMnO}_{4}$ with an oxalate in acidic medium, $\mathrm{MnO}_{4}^{-}$ is reduced to $\mathrm{Mn}^{2+}$ and $\mathrm{C}_{2} \mathrm{O}_{4}^{2-}$ is oxidised to $\mathrm{CO}_{2}$. Hence, $50 \mathrm{~mL}$ of 0.02 $\mathrm{M} \mathrm{KMnO}_{4}$ is equivalent to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$100 \mathrm{~mL}$ of $0.05 \mathrm{M} \mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$50 \mathrm{~mL}$ of $0.05 \mathrm{M} \mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$25 \mathrm{~mL}$ of $0.2 \mathrm{M} \mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$50 \mathrm{~mL}$ of $0.10 \mathrm{M} \mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$50 \mathrm{~mL}$ of $0.05 \mathrm{M} \mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}$</span> </div>

answer:
<div class="solution">eg. of \(\mathrm{KMnO}_{4}\)
<br/>\(\begin{aligned}
<br/>&amp;=5 \times 0.02 \times 50 \\
<br/>&amp;=250 \times 0.02 \\
<br/>&amp;=509 .
<br/>\end{aligned}\)
<br/><img src="images\hfhyry784783647y2.png"/> <br/>eq. of \(\mathrm{KAnO}_{4} \equiv \mathrm{eq}_{1}\) of \(\mathrm{C}_{2} \mathrm{O}_{4}{ }_{2}-\)
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ewehr3yi48ejw37.png"/> <br/>\(\begin{aligned} 5=\operatorname{option} 2 &amp;=50 \times 0.05 \times 2 \\ &amp;=589 . \end{aligned}\)</div>


tags: ['Chemistry', 'Chemical Kinetics', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
in-the-reaction-p-q-r-s-the-time-taken-for-75-reaction-of-p-is-twice-the-time-taken-for-50-reaction-of-p-the-concentration-of-q-varies-with-reaction-t-1

slug:
in-the-reaction-p-q-r-s-the-time-taken-for-75-reaction-of-p-is-twice-the-time-taken-for-50-reaction-of-p-the-concentration-of-q-varies-with-reaction-t-1-23306

question:
<div class="question">In the reaction, $\mathrm{P}+\mathrm{Q} \longrightarrow ? \mathrm{R}+\mathrm{S}$
<br/> <br/>The time taken for $75 \%$ reaction of $P$ is twice the time taken for $50 \%$ reaction of $\mathrm{P}$. The concentration of $\mathrm{Q}$ varies with reaction time as shown in the figure. The overall order of the reaction is
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/3_u8hm7FhGwM0uB__UbzDnr-Oa_Dr56KnaDRGDKlxKM.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">For $\mathrm{P}$, if $\mathrm{t}_{50 \%}=\mathrm{x}$ then $\mathrm{t}_{75 \%}=2 \mathrm{x}$
<br/> <br/>This is true only for first order reaction. So, order with respect to $\mathrm{P}$ is $1 .$ Further the graph shows that concentration of $\mathrm{Q}$ decreases with time. So rate, with respect to $\mathrm{Q},$ remains constant. Hence, it is zero order wrt Q. So, overall order is $1+0=1$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
title:
in-the-reaction-p-q-r-s-the-time-taken-for-75-reaction-of-p-is-twice-the-time-taken-for-50-reaction-of-p-the-concentration-of-q-varies-with-reaction-t-2

slug:
in-the-reaction-p-q-r-s-the-time-taken-for-75-reaction-of-p-is-twice-the-time-taken-for-50-reaction-of-p-the-concentration-of-q-varies-with-reaction-t-2-18243

question:
<div class="question">In the reaction, $\mathrm{P}+\mathrm{Q} \longrightarrow ? \mathrm{R}+\mathrm{S}$
<br/> <br/>The time taken for $75 \%$ reaction of $P$ is twice the time taken for $50 \%$ reaction of $\mathrm{P}$. The concentration of $\mathrm{Q}$ varies with reaction time as shown in the figure. The overall order of the reaction is
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/3_u8hm7FhGwM0uB__UbzDnr-Oa_Dr56KnaDRGDKlxKM.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">For $\mathrm{P}$, if $\mathrm{t}_{50 \%}=\mathrm{x}$ then $\mathrm{t}_{75 \%}=2 \mathrm{x}$
<br/> <br/>This is true only for first order reaction. So, order with respect to $\mathrm{P}$ is $1 .$ Further the graph shows that concentration of $\mathrm{Q}$ decreases with time. So rate, with respect to $\mathrm{Q},$ remains constant. Hence, it is zero order wrt Q. So, overall order is $1+0=1$</div>


title: in-the-reaction-which-of-the-following-compounds-will-be-formed
slug: in-the-reaction-which-of-the-following-compounds-will-be-formed-76739
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the reaction:
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/3sQUFeAzhZ-x2HYLmh2oKGFgkxcUPeirztxttUDe2pI.original.fullsize.png"/><br/> <br/> <br/>Which of the following compounds will be formed?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/55bFWaN1hiu3uXPGdkq1P849MKX0r3VrFk9iG4A8JDY.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/-30-L16hbrVodBUm9NbOxLOosv0CVHaPTnBCMAF1Wjk.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/zH2MpGqiqmQEBYo98KovRuRJpj6OYZ-Rf-WCbke5UF8.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/FvB-QHYmj6kbVLO6I4r2F0P_I7bckQ15ymjQ1NhXR_o.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/-30-L16hbrVodBUm9NbOxLOosv0CVHaPTnBCMAF1Wjk.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/jeb0v30uIPQXPk6ymp3tyyigOGvFEB3DZZM-BsqufZo.original.fullsize.png"/></div>


title: in-the-reaction-x-and-y-are-respectively-1
slug: in-the-reaction-x-and-y-are-respectively-1-89832
tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question"><p>In the reaction <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">X</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">Y</mi></mfenced></math> are respectively.</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/e84e7933-f335-46c9-ba6c-e10d6c38ff10-4c85ae67-614f-4990-a868-28821819c8d3-image.png" style="width: 677px; height: 163px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img height="179" src="https://cdn-question-pool.getmarks.app/pyq/bitsat/58825c06-434a-4693-9333-1ad3c3914bbf-image.png" width="272"/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img height="169" src="https://cdn-question-pool.getmarks.app/pyq/bitsat/e7f0f7d7-c00b-497a-877e-356e49cbfc65-image.png" width="254"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/c2536ab8-2613-42a3-9a44-8968003c8c45-image.png" style="width: 317px; height: 168px;"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img height="164" src="https://cdn-question-pool.getmarks.app/pyq/bitsat/fbd04f9f-f710-432d-ac26-a479fd2d4dce-image.png" width="343"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img height="169" src="https://cdn-question-pool.getmarks.app/pyq/bitsat/e7f0f7d7-c00b-497a-877e-356e49cbfc65-image.png" width="254"/></span> </div>

answer:
<div class="solution"><p>The given reaction is for preparation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>DDT</mi></math> (dichlorodiphenyl trichloroethane). Complete reaction is given as follows.</p><p>(i)</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/c49419d1-5434-42c7-b515-e22645f3c73a-image.png" style="width: 300px; height: 130px;"/></p><p>(ii) <img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/230d6a1e-e8e5-4e0d-9c43-c0793d7fc9ca-image.png" style="width: 500px; height: 97px;"/></p><p>Dichloro-diphenyl-trichloroethane (DDT) is prepared by heating chloral and chlorobenzene in 1:2 ratio in the presence of conc. sulphuric acid.</p><p>DDT is used as an insecticide. It is also used as a mosquito repellant.</p></div>


title: in-the-real-number-system-the-equation-x-3-4-x-1-x-8-6-x-1-1-has-1
slug: in-the-real-number-system-the-equation-x-3-4-x-1-x-8-6-x-1-1-has-1-96766
tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">In the real number system, the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>x</mi><mo>+</mo><mn>3</mn><mo>-</mo><mn>4</mn><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt></msqrt><mo>+</mo><msqrt><mi>x</mi><mo>+</mo><mn>8</mn><mo>-</mo><mn>6</mn><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt></msqrt><mo>=</mo><mn>1</mn></math> has</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">No solution</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Exactly two distinct solution</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Exactly four distinct solution</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Infinitely many solution</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Infinitely many solution</span> </div>

answer:
<div class="solution"><p>We have,<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>x</mi><mo>+</mo><mn>3</mn><mo>-</mo><mn>4</mn><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt></msqrt><mo>+</mo><msqrt><mi>x</mi><mo>+</mo><mn>8</mn><mo>-</mo><mn>6</mn><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt></msqrt><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mo>(</mo><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><msup><mo>)</mo><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>×</mo><mn>2</mn><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><mo>+</mo><mn>4</mn></msqrt><mo>+</mo><msqrt><mo>(</mo><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><msup><mo>)</mo><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>×</mo><mn>3</mn><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><mo>+</mo><mn>9</mn></msqrt><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msqrt><mo>(</mo><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><mo>-</mo><mn>2</mn><msup><mo>)</mo><mn>2</mn></msup></msqrt><mo>+</mo><msqrt><mo>(</mo><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><mo>-</mo><mn>3</mn><msup><mo>)</mo><mn>2</mn></msup></msqrt><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mo>|</mo><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><mo>-</mo><mn>2</mn><mo>|</mo><mo>+</mo><mo>|</mo><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><mo>-</mo><mn>3</mn><mo>|</mo><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mo>|</mo><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><mo>-</mo><mn>2</mn><mo>|</mo><mo>+</mo><mo>|</mo><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>1</mn><mo>-</mo><mn>3</mn><mo>|</mo><mo>=</mo><mfenced close="|" open="|"><mrow><mo>(</mo><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><mo>-</mo><mn>2</mn><mo>)</mo><mo>-</mo><mo>(</mo><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><mo>-</mo><mn>3</mn><mo>)</mo></mrow></mfenced></math></p><p>We know that, </p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>x</mi><mo>-</mo><mi>b</mi><mo>|</mo><mo>=</mo><mfenced close="|" open="|"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>)</mo><mo>-</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>b</mi><mo>)</mo></mrow></mfenced></math></p><p>then <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>b</mi><mo>)</mo><mo>&lt;</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo>(</mo><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><mo>-</mo><mn>2</mn><mo>)</mo><mo>(</mo><msqrt><mi>x</mi><mo>-</mo><mn>1</mn></msqrt><mo>-</mo><mn>3</mn><mo>)</mo><mo>&lt;</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>2</mn><mo>&lt;</mo><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>1</mn><mo>&lt;</mo><mn>3</mn><mo>⇒</mo><mn>5</mn><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mn>10</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Equation has infinite many solutions.</p></div>


tags: ['Physics', 'Units and Dimensions', 'BITSAT', 'BITSAT 2009']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P1.db
title:
in-the-relation-p-is-pressure-z-is-distance-k-is-boltzmann-constant-and-is-the-temperature-the-dimensional-formula-of-will-be

slug:
in-the-relation-p-is-pressure-z-is-distance-k-is-boltzmann-constant-and-is-the-temperature-the-dimensional-formula-of-will-be-56074

question:
<div class="question">In the relation:<br/>$\mathrm{P}$ is pressure, $\mathrm{Z}$ is distance, $\mathrm{k}$ is Boltzmann constant and $\theta$ is the temperature. The<br/>dimensional formula of $\beta$ will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{0}\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{1}\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-1}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{-1}\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{0}\right]$</span> </div>


tags: ['Physics', 'Experimental Physics', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$x$</span> </div>
source_db: MarksBatch2_P1.db
title:
in-the-shown-arrangement-of-the-experiment-of-the-meter-bridge-if-a-c-corresponding-to-null-deflection-of-galvanometer-is-x-what-would-be-its-value-if

slug:
in-the-shown-arrangement-of-the-experiment-of-the-meter-bridge-if-a-c-corresponding-to-null-deflection-of-galvanometer-is-x-what-would-be-its-value-if-47383

question:
<div class="question">In the shown arrangement of the experiment of the meter bridge if $A C$ corresponding to null deflection of galvanometer is $\mathrm{x},$ what would be its value if the radius of the wire $\mathrm{AB}$ is doubled?
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/6MTrwJR0Q9U6cIcG-3yKIabRfQqq2XD5LiPnycjw120.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x / 4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 x$</span> </li> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/OqbZXdYbXXHnqfLVcxwLEePOLZcLRysHouszNpdF9gU.original.fullsize.png"/><br/> <br/> <br/>At null point $\frac{R_{1}}{R_{2}}=\frac{R_{3}}{R_{4}}=\frac{x}{100-x}$
<br/> <br/>If radius of the wire is doubled, then the resistance of $A C$ will change and also the resistance of $C B$ will change. But since
<br/> <br/>$\frac{R_{1}}{R_{2}}$ does not change so, $\frac{R_{3}}{R_{4}}$ should also not change at null point. Therefore the point $C$ does not change.</div>


title: in-the-shown-situation-if-middle-portion-of-the-lens-is-painted-black-then
slug: in-the-shown-situation-if-middle-portion-of-the-lens-is-painted-black-then-77017
tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question"><p>In the shown situation, if middle portion of the lens is painted black, then</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/88a8e0cf-5050-4907-8217-10ab27ba46a8-image.png" style="width: 260px; height: 152px;"/></p></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">complete image of the object will form with low intensity</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">complete image of the object will form with high intensity</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">incomplete image of the object will form with low intensity</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">incomplete image of the object will form with high intensity</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">complete image of the object will form with low intensity</span> </div>

answer:
<div class="solution">If middle portion of lens is painted black, then less number of refracted rays will intersect, therefore intensity of the corresponding image of object will be less. Though, the rays coming from object refracted with small portion of lens, but complete image of object will be formed but of lesser intensity.</div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
title:
in-the-youngs-doubleslit-experiment-the-intensity-of-light-at-a-point-on-the-screen-where-the-path-difference-is-is-k-being-the-wave-length-of-light-u

slug:
in-the-youngs-doubleslit-experiment-the-intensity-of-light-at-a-point-on-the-screen-where-the-path-difference-is-is-k-being-the-wave-length-of-light-u-64304

question:
<div class="question">In the Young's double-slit experiment, the intensity of light at a point on the screen where the path difference is $\lambda$ is $\mathrm{K},(\lambda$ being the wave length of light used). The intensity at a point where the path difference is $\lambda / 4,$ will be :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{K}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{K} / 4$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{K} / 2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Zero</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{K} / 2$</span> </div>

answer:
<div class="solution">For path difference $\lambda,$ phase difference $=2 \pi \mathrm{rad}$
<br/> <br/>For path difference $\frac{\lambda}{4},$ phase difference
<br/> <br/>$$
<br/> <br/>=\frac{\pi}{2} \mathrm{rad}
<br/> <br/>$$
<br/> <br/>As $\mathrm{K}=4 \mathrm{I}_{0}$ so intensity at given point where
<br/> <br/>path difference is $\frac{\lambda}{4}$
<br/> <br/>$\mathrm{K}^{\prime}=4 \mathrm{I}_{0} \cos ^{2}\left(\frac{\pi}{4}\right)\left(\cos \frac{\pi}{4}=\cos 45^{\circ}\right)$
<br/> <br/>$=2 \mathrm{I}_{0}=\frac{\mathrm{K}}{2}$</div>


tags: ['Chemistry', 'States of Matter', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
title:
in-van-der-waals-equation-of-state-for-a-nonideal-gas-the-term-that-accounts-for-intermolecular-forces-is

slug:
in-van-der-waals-equation-of-state-for-a-nonideal-gas-the-term-that-accounts-for-intermolecular-forces-is-15953

question:
<div class="question">In van der Waal's equation of state for a nonideal gas, the term that accounts for intermolecular forces is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(V-b)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$R T$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(P+\frac{a}{V^2}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(R T)^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(P+\frac{a}{V^2}\right)$</span> </div>

answer:
<div class="solution">$\left(P+\frac{a}{V^2}\right)(V-b)=R T$; Here $\left(P+\frac{a}{V^2}\right)$ represents the intermolecular forces.</div>


title: in-which-case-the-no-2-will-attack-at-the-meta-position-1
slug: in-which-case-the-no-2-will-attack-at-the-meta-position-1-90329
tags: ['Chemistry', 'Hydrocarbons', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">In which case the $\mathrm{NO}_{2}$ will attack at the meta position
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/B08WC36JMvrkSIH1PAFRv9JK32u6aNY75bfiPSlSH-A.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">I, II, III</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">II, IV</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">II and III only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">II only</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">I, II, III</span> </div>

answer:
<div class="solution">$-\mathrm{CCl}_{3},-\mathrm{NO}_{2}$ and $-\mathrm{NH}_{3}$ are
<br/>$m$ -directing in nature.</div>


title: in-which-of-the-following-conversions-phosphorus-pentachloride-is-used-as-the-reagent
slug: in-which-of-the-following-conversions-phosphorus-pentachloride-is-used-as-the-reagent-20523
tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
question:
<div class="question">In which of the following conversions, phosphorus pentachloride is used as the reagent?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{H}_{2} \mathrm{C}=\mathrm{CH}_{2} \longrightarrow \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Cl}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH} \longrightarrow \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Cl}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{H}_{3} \mathrm{C}-\mathrm{O}-\mathrm{CH}_{3} \longrightarrow \mathrm{CH}_{3} \mathrm{Cl}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{CH} \equiv \mathrm{CH} \longrightarrow \mathrm{CH}_{2}=\mathrm{CHCl}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH} \longrightarrow \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Cl}$</span> </div>

answer:
<div class="solution">When ethyl alcohol is treated with $\mathrm{PCl}_{5}$, then ethyl chloride is formed.<br/>$$<br/>\begin{aligned}<br/>\mathrm{CH}_{3} \mathrm{CH}_{2}-\mathrm{OH} &amp;+\mathrm{PCl}_{5} \longrightarrow_{\Delta} \\<br/>&amp; \mathrm{CH}_{3} \mathrm{CH}_{2}-\mathrm{Cl}+\mathrm{HCl}+\mathrm{POCl}_{3}<br/>\end{aligned}<br/>$$</div>


title: in-which-of-the-following-reactions-h-2-o-2-is-acting-as-a-reducing-agent
slug: in-which-of-the-following-reactions-h-2-o-2-is-acting-as-a-reducing-agent-81669
tags: ['Chemistry', 'Redox Reactions', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
question:
<div class="question">In which of the following reactions, $\mathrm{H}_{2} \mathrm{O}_{2}$ is acting as a reducing agent?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{H}_{2} \mathrm{O}_{2}+\mathrm{SO}_{2} \rightarrow \mathrm{H}_{2} \mathrm{SO}_{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{KI}+\mathrm{H}_{2} \mathrm{O}_{2} \rightarrow 2 \mathrm{KOH}+\mathrm{I}_{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad \mathrm{PbS}+4 \mathrm{H}_{2} \mathrm{O}_{2} \rightarrow \mathrm{PbSO}_{4}+4 \mathrm{H}_{2} \mathrm{O}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{Ag}_{2} \mathrm{O}+\mathrm{H}_{2} \mathrm{O}_{2} \rightarrow 2 \mathrm{Ag}+\mathrm{H}_{2} \mathrm{O}+\mathrm{O}_{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Ag}_{2} \mathrm{O}+\mathrm{H}_{2} \mathrm{O}_{2} \rightarrow 2 \mathrm{Ag}+\mathrm{H}_{2} \mathrm{O}+\mathrm{O}_{2}$</span> </div>

answer:
<div class="solution">$\mathrm{SO}_{2}$ changes to $\mathrm{H}_{2} \mathrm{SO}_{4}$ (O.N. changes from +4 to +6 oxidation $)$ $2 \mathrm{KI} \rightarrow \mathrm{I}_{2}(\mathrm{O} . \mathrm{S} .$ changes from -1 to 0<br/>oxidation) $\mathrm{PbS} \rightarrow \mathrm{PbSO}_{4}(\mathrm{O} . \mathrm{S}$. changes from -2 to +6<br/>oxidation) $\mathrm{Ag}_{2} \mathrm{O} \rightarrow 2 \mathrm{Ag}(\mathrm{O} . \mathrm{S} .$ changes from +1 to 0<br/>reduction)</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
title:
in-which-of-the-following-reactions-standard-entropy-change-s-is-positive-and-standard-gibbs-energy-change-g-decreases-sharply-with-increasing-tempera

slug:
in-which-of-the-following-reactions-standard-entropy-change-s-is-positive-and-standard-gibbs-energy-change-g-decreases-sharply-with-increasing-tempera-42406

question:
<div class="question">In which of the following reactions, standard entropy change $\left(\Delta \mathrm{S}^{\circ}\right)$ is positive and standard Gibb's energy change $\left(\Delta \mathrm{G}^{\circ}\right)$ decreases sharply with increasing temperature?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{C}$ (graphite) $+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}(\mathrm{g})$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{CO}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g})$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{Mg}(\mathrm{s})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{MgO}(\mathrm{s})$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2} \mathrm{C}$ (graphite) $+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \frac{1}{2} \mathrm{CO}_{2}(\mathrm{~g})$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{C}$ (graphite) $+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}(\mathrm{g})$</span> </div>

answer:
<div class="solution">Since, in the first reaction gaseous products are forming from solid carbon hence entropy will increase i.e. $\Delta \mathrm{S}=+\mathrm{ve}$.
<br/>$\mathrm{C}(\mathrm{gr} .)+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}(\mathrm{g}) ; \Delta \mathrm{S}^{\circ}=+\mathrm{ve}$
<br/>Since, $\Delta \overrightarrow{\mathrm{G}^{\circ}}=\Delta \mathrm{H}^{\circ}-\mathrm{T} \Delta \mathrm{S}$ hence the value of
<br/>$\Delta$ G decrease on increasing temperature.</div>


tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
in-which-pair-or-pairs-is-the-stronger-bond-found-in-the-first-species-i-o-2-2-o-2-ii-n-2-n-2-iii-no-no

slug:
in-which-pair-or-pairs-is-the-stronger-bond-found-in-the-first-species-i-o-2-2-o-2-ii-n-2-n-2-iii-no-no-16055

question:
<div class="question">In which pair or pairs is the stronger bond found in the first species?
<br/>I. $\mathrm{O}_2^{2-}, \mathrm{O}_2$;           II. $\mathrm{N}_2, \mathrm{~N}_2^{+}$ III. $\mathrm{NO}^{+}, \mathrm{NO}^{-}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>I only
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>II only
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>I and II only
<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"> <br/>II and III only</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>II and III only</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ekQnbHmAdXQXv6IoU5trsD3ixQ9-xL9Byqgtb06Si28.original.fullsize.png"/><br/> <br/>Larger the bond order, stronger the bond formed.</div>


title: in-xef-2-xef-4-xef-6-the-number-of-lone-pairs-on-xe-are-respectively-1
slug: in-xef-2-xef-4-xef-6-the-number-of-lone-pairs-on-xe-are-respectively-1-19519
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
question:
<div class="question">In $\mathrm{XeF}_{2}, \mathrm{XeF}_{4}, \mathrm{XeF}_{6}$ the number of lone pairs on $\mathrm{Xe}$ are respectively</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2,3,1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1,2,3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4,1,2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$3,2,1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3,2,1$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/poozu7ppK0XX5QeghKr0J1d8mTTY5QRhxzyKCkmrntU.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/QVhmwKY6fzOtwbBnLk6midAtZBAPqXmSKi0flSqBDCE.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/gp5Xq268TgoDT2CjxBP_XNLH0OMIYpsTy0KE2sMXMdc.original.fullsize.png"/></div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">14 th</span> </div>
answer: <div class="solution">$\mathrm{n}_{1} \lambda_{1}=\mathrm{n}_{2} \lambda_{2}$</div>
source_db: MarksBatch2_P1.db
title:
in-youngs-double-slit-experiment-10-th-order-maximum-is-obtained-at-the-point-of-observation-in-the-interference-pattern-for-7000-a-if-the-source-is-r

slug:
in-youngs-double-slit-experiment-10-th-order-maximum-is-obtained-at-the-point-of-observation-in-the-interference-pattern-for-7000-a-if-the-source-is-r-95665

question:
<div class="question">In Young's double slit experiment 10 th order maximum is obtained at the point of observation in the interference pattern for $\lambda=7000 \AA$. If the source is replaced by another one of wavelength $5000 \AA$ then the order of maximum at the same point will be-</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">12 th</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">14 th</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">16 th</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">18 th</span> </li> </ul>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1: 9</span> </div>
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P1.db
title:
in-youngs-double-slit-experiment-if-the-slit-widths-are-in-the-ratio-1-2-the-ratio-of-the-intensities-at-minima-and-maxima-will-be

slug:
in-youngs-double-slit-experiment-if-the-slit-widths-are-in-the-ratio-1-2-the-ratio-of-the-intensities-at-minima-and-maxima-will-be-86755

question:
<div class="question">In Young's double slit experiment, if the slit widths are in the ratio $1: 2,$ the ratio of the intensities at minima and maxima will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1: 2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1: 3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1: 4</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1: 9</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
in-youngs-double-slit-experiment-intensity-at-a-point-is-1-4-of-the-maximum-intensity-angular-position-of-this-point-is

slug:
in-youngs-double-slit-experiment-intensity-at-a-point-is-1-4-of-the-maximum-intensity-angular-position-of-this-point-is-21133

question:
<div class="question">In Young's double slit experiment, intensity at a point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced></math> of the maximum intensity. Angular position of this point is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mi>λ</mi><mi>d</mi></mfrac></mfenced></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mi>λ</mi><mrow><mn>2</mn><mi>d</mi></mrow></mfrac></mfenced></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mi>λ</mi><mrow><mn>3</mn><mi>d</mi></mrow></mfrac></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mi>λ</mi><mrow><mn>4</mn><mi>d</mi></mrow></mfrac></mfenced></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mi>λ</mi><mrow><mn>3</mn><mi>d</mi></mrow></mfrac></mfenced></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><msub><mi>I</mi><mi>max</mi></msub><msup><mi>cos</mi><mn>2</mn></msup><mfenced><mfrac><mi>ϕ</mi><mn>2</mn></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>I</mi><mi>max</mi></msub><mn>4</mn></mfrac><mo>=</mo><msub><mi>I</mi><mi>max</mi></msub><msup><mi>cos</mi><mn>2</mn></msup><mfrac><mi>ϕ</mi><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfrac><mi>ϕ</mi><mn>2</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>ϕ</mi><mn>2</mn></mfrac><mo>=</mo><mfrac><mi>π</mi><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>ϕ</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mn>3</mn></mfrac><mo>=</mo><mfenced><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>λ</mi></mfrac></mfenced><mo>×</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>=</mo><mi>d</mi><mi>sin</mi><mi>θ</mi></math></p><p>Putting the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mi>x</mi></math> in Eq. (i), we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mi>θ</mi><mo>=</mo><mfrac><mi>λ</mi><mrow><mn>3</mn><mi>d</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mi>λ</mi><mrow><mn>3</mn><mi>d</mi></mrow></mfrac></mfenced></math></p></div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
title:
in-youngs-double-slit-experiment-the-slits-are-2-mm-apart-and-are-illuminated-by-photons-of-two-wavelengths-1-12000-and-2-10000-at-what-minimum-distan

slug:
in-youngs-double-slit-experiment-the-slits-are-2-mm-apart-and-are-illuminated-by-photons-of-two-wavelengths-1-12000-and-2-10000-at-what-minimum-distan-52063

question:
<div class="question">In Young's double slit experiment, the slits are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>mm</mi></math> apart and are illuminated by photons of two wavelengths <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub><mo>=</mo><mn>12000</mn><mo>Å</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>2</mn></msub><mo>=</mo><mn>10000</mn><mo>Å </mo></math>. At what minimum distance from the common central bright fringe on the screen <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">m</mi></math> from the slit will a bright fringe from one interference pattern coincide with a bright fringe from the other?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mi>mm</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mi>mm</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi>mm</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi>mm</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mi>mm</mi></math></span> </div>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub><mo>=</mo><mn>12000</mn><mo>Å </mo></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>2</mn></msub><mo>=</mo><mn>10000</mn><mo>Å </mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mn>2</mn><mo> </mo><mi>cm</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>2</mn><mo> </mo><mi>mm</mi><mo>=</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></math></p><p>We have, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>λ</mi><mn>1</mn></msub><msub><mi>λ</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>n</mi><mn>2</mn></msub><msub><mi>n</mi><mn>1</mn></msub></mfrac><mo>=</mo><mfrac><mn>12000</mn><mn>10000</mn></mfrac><mo>=</mo><mfrac><mn>6</mn><mn>5</mn></mfrac></math></p><p>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mfrac><mrow><msub><mi>n</mi><mn>1</mn></msub><msub><mi>λ</mi><mn>1</mn></msub><mi>D</mi></mrow><mi>d</mi></mfrac></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>n</mi><mn>1</mn></msub><mo>=</mo><mn>5</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>5</mn><mo>×</mo><mn>12000</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>10</mn></mrow></msup><mo>×</mo><mn>2</mn></mrow><mrow><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>5</mn><mo>×</mo><mn>1</mn><mo>.</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>10</mn></mrow></msup><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo>=</mo><mn>6</mn><mo> </mo><mi>mm</mi></math></p></div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
in-youngs-doubleslit-experiment-the-distance-between-slits-and-screen-is-2-m-and-distance-between-slits-is-0-25-mm-a-light-of-wavelength-800-nm-is-use

slug:
in-youngs-doubleslit-experiment-the-distance-between-slits-and-screen-is-2-m-and-distance-between-slits-is-0-25-mm-a-light-of-wavelength-800-nm-is-use-67961

question:
<div class="question">In Young's double-slit experiment, the distance between slits and screen is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">m</mi></math> and distance between slits is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>25</mn><mo> </mo><mi>mm</mi></math>. A light of wavelength <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>800</mn><mo> </mo><mi>nm</mi></math> is used to find fringes on the screen. If the screen moves with a speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, then first maxima will move with a speed of:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mo> </mo><mi>mm</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mi>mm</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mi>mm</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn><mo> </mo><mi>mm</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mo> </mo><mi>mm</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </div>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mn>2</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>,</mo><mo> </mo><mi>d</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>25</mn><mo> </mo><mi>mm</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></math></p><p>Velocity of screen, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>D</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mn>5</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Speed of first maxima <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi>β</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo>?</mo></math></p><p>In Young's double-slit experiment, fringe width <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mfrac><mrow><mi>D</mi><mi>λ</mi></mrow><mi>d</mi></mfrac></math></p><p>On differentiating with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>β</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mi>λ</mi><mi>d</mi></mfrac><mo>·</mo><mfrac><mrow><mi>d</mi><mi>D</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>800</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>9</mn></mrow></msup></mrow><mrow><mn>2</mn><mo>.</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup></mrow></mfrac><mo>×</mo><mn>5</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>16</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>16</mn><mo> </mo><mi>mm</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p></div>


title: increasing-order-of-rms-velocities-of-h-2-o-2-n-2-and-hbr-is
slug: increasing-order-of-rms-velocities-of-h-2-o-2-n-2-and-hbr-is-72711
tags: ['Chemistry', 'States of Matter', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
question:
<div class="question">Increasing order of rms velocities of $\mathrm{H}_{2}, \mathrm{O}_{2}, \mathrm{~N}_{2}$ and $\mathrm{HBr}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{H}_{2}&gt;\mathrm{O}_{2}&gt;\mathrm{N}_{2}&gt;\mathrm{HBr}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{HBr}&lt;\mathrm{O}_{2}&lt;\mathrm{N}_{2}&lt;\mathrm{H}_{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{H}_{2}&gt;\mathrm{N}_{2}&lt;\mathrm{O}_{2}&gt;\mathrm{HBr}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{HBr}&gt;\mathrm{N}_{2}&lt;\mathrm{O}_{2}&lt;\mathrm{H}_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{HBr}&lt;\mathrm{O}_{2}&lt;\mathrm{N}_{2}&lt;\mathrm{H}_{2}$</span> </div>

answer:
<div class="solution">RMS velocity of molecules depends on mass. If mol. wt. increases, rms velocity of melocules decreases. $\mathrm{rms} \propto \frac{1}{\sqrt{\mathrm{m} \cdot \mathrm{wt}}}$
<br/> <br/>The order of increasing $\mathrm{m} . \mathrm{wt}$. is
<br/> <br/>$\mathrm{H}_{2}&lt;\mathrm{N}_{2}&lt;\mathrm{O}_{2}&lt;\mathrm{HBr}$
<br/> <br/>Order of $V_{\mathrm{mx}}$ of molecules. $\mathrm{HBr}&lt;\mathrm{O}_{2}&lt;\mathrm{N}_{2}&lt;\mathrm{H}_{2}$</div>


tags: ['Chemistry', 'Chemistry in Everyday Life', 'BITSAT', 'BITSAT 2013']
answer: <div class="solution">Insulin is a biochemically active peptide harmone secreted by pancreas.</div>
source_db: MarksBatch2_P1.db
title:
insulin-production-and-its-action-in-human-body-are-responsible-for-the-level-of-diabetes-this-compound-belongs-to-which-of-the-following-categories-1

slug:
insulin-production-and-its-action-in-human-body-are-responsible-for-the-level-of-diabetes-this-compound-belongs-to-which-of-the-following-categories-1-67950

question:
<div class="question">Insulin production and its action in human body are responsible for the level of diabetes. This compound belongs to which of the following categories?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">An enzyme</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">A hormone</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">A co-enzyme</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">An antibiotic</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">A hormone</span> </div>


title: int-4-cos-leftxfracpi6right-cos-2-x-cdot-cos-leftfrac5-pi6xright-d-x
slug: int-4-cos-leftxfracpi6right-cos-2-x-cdot-cos-leftfrac5-pi6xright-d-x-33451
tags: ['Mathematics', 'Definite Integration', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
question:
<div class="question">\(\int 4 \cos \left(x+\frac{\pi}{6}\right) \cos 2 x \cdot \cos \left(\frac{5 \pi}{6}+x\right) d x\)</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(-\left(x+\frac{\sin 4 x}{4}+\frac{\sin 2 x}{2}\right)+c\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(-\left(x+\frac{\sin 4 x}{4}-\frac{\sin 2 x}{2}\right)+c\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(-\left(x-\frac{\sin 4 x}{4}+\frac{\sin 2 x}{2}\right)+c\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(-\left(x-\frac{\sin 4 x}{4}+\frac{\cos 2 x}{2}\right)+c\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(-\left(x+\frac{\sin 4 x}{4}+\frac{\sin 2 x}{2}\right)+c\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \int 4 \cos \left(x+\frac{\pi}{6}\right) \cos 2 x \cdot \cos \left(\frac{5 \pi}{6}+x\right) d x \\<br/>&amp; =2 \int\left(\cos (2 x+\pi) \cos \frac{2 \pi}{3}\right) \cos 2 x d x \\<br/>&amp; =2 \int\left(-\cos 2 x-\frac{1}{2}\right) \cos 2 x d x \\<br/>&amp; =\int\left(-2 \cos ^2 2 x-\cos 2 x\right) d x \\<br/>&amp; =-\int(1+\cos 4 x+\cos 2 x) d x \\<br/>&amp; =-x-\frac{\sin 4 x}{x}-\frac{\sin 2 x}{2}+c<br/>\end{aligned}\)</div>


title: interparticle-forces-present-in-nylon-66-are
slug: interparticle-forces-present-in-nylon-66-are-57333
question: <div class="question">Interparticle forces present in nylon 66 are</div>
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'BITSAT', 'BITSAT 2009']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">van der Waal's</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">hydrogen bonding</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">dipole-dipole interactions</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">hydrogen bonding</span> </div>


tags: ['Mathematics', 'Mathematical Reasoning', 'BITSAT', 'BITSAT 2013']
answer: <div class="solution">$\mathrm{T} \mathrm{T} \mathrm{T} \mathrm{T}$</div>
source_db: MarksBatch2_P1.db
title:
inthetruth-table-for-the-statement-p-q-q-p-the-last-column-has-the-truth-value-in-the-following-order-is

slug:
inthetruth-table-for-the-statement-p-q-q-p-the-last-column-has-the-truth-value-in-the-following-order-is-12879

question:
<div class="question">Inthetruth table for the statement $(p \wedge q) \rightarrow(q \vee \sim p)$, the last column has the truth value in the following order is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">TTFF</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">FTTT</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">TFTT</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{TTTT}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{TTTT}$</span> </div>


title: ionic-hydrides-reacts-with-water-to-give
slug: ionic-hydrides-reacts-with-water-to-give-41601
question: <div class="question">Ionic hydrides reacts with water to give</div>
tags: ['Chemistry', 'Hydrogen', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">acidic solutions</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">hydride ions</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">basic solutions</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">electrons</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">basic solutions</span> </div>

answer:
<div class="solution">Ionic hydrides give the basic solution when it reacts with water, e.g.,<br/>$\mathrm{LiH}+\mathrm{H}_2 \mathrm{O} \longrightarrow \mathrm{LiOH}+\mathrm{H}_2$</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
title:
it-is-given-that-the-events-a-and-b-are-such-that-pafrac14-pa-mid-bfrac12-quad-and-quad-pb-mid-afrac23-then-pb-is

slug:
it-is-given-that-the-events-a-and-b-are-such-that-pafrac14-pa-mid-bfrac12-quad-and-quad-pb-mid-afrac23-then-pb-is-54677

question:
<div class="question">It is given that the events \(A\) and \(B\) are such that \(P(A)=\frac{1}{4}, P(A \mid B)=\frac{1}{2} \quad\) and \(\quad P(B \mid A)=\frac{2}{3}\). Then \(P(B)\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{1}{6}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{1}{3}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{2}{3}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{1}{2}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{1}{3}\)</span> </div>

answer:
<div class="solution">\(P(A)=1 / 4, P(A / B)=\frac{1}{2}, P(B / A)=2 / 3\)<br/>By conditional probability,<br/>\(\begin{aligned}<br/>&amp; P(A \cap B)=P(A) P(B / A)=P(B) P(A / B) \\<br/>&amp; \Rightarrow \frac{1}{4} \times \frac{2}{3}=P(B) \times \frac{1}{2} \Rightarrow P(B)=\frac{1}{3}<br/>\end{aligned}\)</div>


title: k-4-fe-cn-6-is-used-in-detecting
slug: k-4-fe-cn-6-is-used-in-detecting-56357
tags: ['Chemistry', 'Practical Chemistry', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">$\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]$ is used in detecting.</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{Fe}^{3+} \mathrm{ion}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{Cu}^{+} \mathrm{ion}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{Cu}^{3+}$ ion</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Fe}^{2+}$ ion</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Fe}^{3+} \mathrm{ion}$</span> </div>

answer:
<div class="solution">$\mathrm{Fe}^{3+}$ ion can be detected by $\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]$
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>4 \mathrm{Fe}^{3+}+3 \mathrm{~K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right] \\
<br/> <br/>&amp; \longrightarrow \mathrm{Fe}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]_{3}+12 \mathrm{~K}^{+}
<br/> <br/>\end{aligned}
<br/> <br/>$$</div>


tags: ['Chemistry', 'Ionic Equilibrium', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
title:
k-c-for-the-the-reaction-ag-cn-2-ag-2-cn-the-equillibrium-constant-at-2-5-c-is-40-1-0-19-then-the-silver-ion-concentration-in-a-solution-which-was-ori

slug:
k-c-for-the-the-reaction-ag-cn-2-ag-2-cn-the-equillibrium-constant-at-2-5-c-is-40-1-0-19-then-the-silver-ion-concentration-in-a-solution-which-was-ori-89845

question:
<div class="question">$K_{c}$ for the the reaction, $\left[\mathrm{Ag}(\mathrm{CN})_{2}\right] \rightleftharpoons \mathrm{Ag}^{+}+2 \mathrm{CN}^{-},$
<br/> <br/>the equillibrium constant at $25^{\circ} \mathrm{C}$ is $4.0 \times 10^{-19}$ then the silver ion concentration in a solution which was originally 0.1 molar in $\mathrm{KCN}$ and 0.03 molar in $\mathrm{AgNO}_{3}$ is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$7.5 \times 10^{18}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$7.5 \times 10^{-19}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$7.5 \times 10^{19}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$7.5 \times 10^{-18}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$7.5 \times 10^{-18}$</span> </div>

answer:
<div class="solution">$\left.\begin{array}{c}2 \mathrm{KCN}+\mathrm{AgNO}=\left[\underset{0.1}{0.03}=\left[\begin{array}{c}\mathrm{Ag}(\mathrm{CN})_{2} \\ 0\end{array}\right]^{-}\right. \\ (0.1-0.06) &amp; 0\end{array}\right] \begin{array}{c}\mathrm{KNO}_{3}+\mathrm{K}_{0}^{+} \\ (0.03)\end{array}$
<br/> <br/>$\therefore\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]^{-}=\mathrm{Ag}^{+}+\underset{\mathrm{a}}{0.03} \mathrm{a} \quad 2 \mathrm{CN}$
<br/> <br/>$(0.03-\mathrm{a}) \quad \mathrm{a} .04(\mathrm{left} \mathrm{from} \mathrm{KCN}$
<br/> <br/>$\quad 0.04 \approx 0.04$
<br/> <br/>$\therefore \quad \mathrm{K}_{\mathrm{c}}=4 \times 10^{-19}=\frac{(0.04)^{2} \times \mathrm{a}}{0.03}$
<br/> <br/>$\mathrm{a}=7.5 \times 10^{-18}$</div>


tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
title:
keplers-third-law-states-that-square-of-period-of-revolution-t-of-a-planet-around-the-sun-is-proportional-to-third-power-of-average-distance-r-between

slug:
keplers-third-law-states-that-square-of-period-of-revolution-t-of-a-planet-around-the-sun-is-proportional-to-third-power-of-average-distance-r-between-25342

question:
<div class="question">Kepler's third law states that square of period of revolution (T) of a planet around the sun, is proportional to third power of average distance $\mathrm{r}$
<br/> <br/>between sun and planet i.e. $\mathrm{T}^{2}=\mathrm{Kr}^{3}$ here $\mathrm{K}$ is constant. If the masses of sun and planet are $\mathrm{M}$ and $\mathrm{m}$ respectively then as per Newton's law of gravitation force of attraction between them is
<br/> <br/>$\mathrm{F}=\frac{\mathrm{GMm}}{\mathrm{r}^{2}}$, here $\mathrm{G}$ is gravitational constant. The relation between $\mathrm{G}$ and $\mathrm{K}$ is described as</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{GMK}=4 \pi^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad \mathrm{K}=\mathrm{G}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{K}=\frac{1}{\mathrm{G}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad \mathrm{GK}=4 \pi^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{GMK}=4 \pi^{2}$</span> </div>

answer:
<div class="solution">As we know, orbital speed, $V_{\text {orb }}=\sqrt{\frac{\text { GM }}{\text { r }}}$ 
<br/> <br/>Time period $\mathrm{T}=\frac{2 \pi \mathrm{r}}{\mathrm{v}_{\text {orb }}}=\frac{2 \pi \mathrm{r}}{\sqrt{\mathrm{GM}}} \sqrt{\mathrm{r}}$
<br/> <br/>Squarring both sides,
<br/> <br/>$\mathrm{T}^{2}=\left(\frac{2 \pi \mathrm{r} \sqrt{\mathrm{r}}}{\sqrt{\mathrm{GM}}}\right)^{2}=\frac{4 \pi^{2}}{\mathrm{GM}} \cdot \mathrm{r}^{3}$
<br/> <br/>$\Rightarrow \quad \frac{\mathrm{T}^{2}}{\mathrm{r}^{3}}=\frac{4 \pi^{2}}{\mathrm{GM}}=\mathrm{K} \Rightarrow \mathrm{GMK}=4 \pi^{2}$</div>


title: ketoenol-tautomerism-is-observed-in
slug: ketoenol-tautomerism-is-observed-in-50716
question: <div class="question">Keto-enol tautomerism is observed in</div>
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/PaiZz9RJz3brrzzggM2nvGcFzdaLJ9EvQBwmizuYzUc.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/rajqT5ZVwGk2zcKH0dt_XKifJOGhC8b734RZ7xPotMA.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/9Sc3gmrF8js04MYsVttGwxpbKweSw1Zett4oiiAMcCc.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/rajqT5ZVwGk2zcKH0dt_XKifJOGhC8b734RZ7xPotMA.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/fNLG-orkNxqwV9Upaq_TIyaOCebTkyjQ4onU1s0Vpcw.original.fullsize.png"/></div>


title: kinetic-theory-of-gases-proves-1
slug: kinetic-theory-of-gases-proves-1-86593
question: <div class="question">Kinetic theory of gases proves</div>
tags: ['Chemistry', 'States of Matter', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
answer: <div class="solution">Kinetic theory of gases proves all the given gas laws.</div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">only Boyle's law</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> only Charles' law</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">only Avogadro's law</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">all of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">all of these</span> </div>


tags: ['Chemistry', 'd and f Block Elements', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
title:
knowing-that-the-chemistry-of-lanthanoidsln-is-dominated-by-its-3-oxidation-state-which-of-the-following-statements-is-incorrect

slug:
knowing-that-the-chemistry-of-lanthanoidsln-is-dominated-by-its-3-oxidation-state-which-of-the-following-statements-is-incorrect-41089

question:
<div class="question">Knowing that the chemistry of lanthanoids(Ln) is dominated by its +3 oxidation state, which of the following statements is incorrect?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The ionic size of Ln (III) decrease in general with increasing atomic number</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Ln (III) compounds are generally colourless.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Ln (III) hydroxide are mainly basic in character.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Because of the large size of the Ln (III) ions the bonding in its compounds is predominantly ionic in character.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Ln (III) compounds are generally colourless.</span> </div>

answer:
<div class="solution">Most of the $\mathrm{Ln}^{3+}$ compounds except $\mathrm{La}^{3+}$ and $\mathrm{Lu}^{3+}$ are coloured due to the presence of $f$ -electrons.</div>


title: laughing-gas-is
slug: laughing-gas-is-93701
question: <div class="question">Laughing gas is</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'BITSAT', 'BITSAT 2015']
answer: <div class="solution">Nitrous oxide (i.e, $\mathrm{N}_{2} \mathrm{O}$ ) is the laughing gas.</div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">nitrogen pentoxide</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">nitrous oxide</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad$ nitrogen trioxide</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">nitric oxide</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">nitrous oxide</span> </div>


title: lenzs-law-is-a-consequence-of-the-law-of-conservation-of
slug: lenzs-law-is-a-consequence-of-the-law-of-conservation-of-77009
question: <div class="question">Lenz's law is a consequence of the law of conservation of</div>
tags: ['Physics', 'Electromagnetic Induction', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">energy</span> </div>
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">charge</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">mass</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">energy</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">momentum</span> </li> </ul>


tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
title:
let-1-2-and-1-2-be-the-roots-of-a-x-2-b-x-c-0-and-px-2-qx-r-0-respectively-if-the-system-of-equations-1-y-2-z-0-and-1-y-2-z-0-has-a-nontrivial-solutio

slug:
let-1-2-and-1-2-be-the-roots-of-a-x-2-b-x-c-0-and-px-2-qx-r-0-respectively-if-the-system-of-equations-1-y-2-z-0-and-1-y-2-z-0-has-a-nontrivial-solutio-21548

question:
<div class="question">Let $\alpha_{1}, \alpha_{2}$ and $\beta_{1}, \beta_{2}$ be the roots of $a x^{2}+b x+c$ $=0$ and $\mathrm{px}^{2}+\mathrm{qx}+\mathrm{r}=0$ respectively. If the system of equations $\alpha_{1} \mathrm{y}+\alpha_{2} \mathrm{z}=0$ and $\beta_{1} \mathrm{y}+\beta_{2} \mathrm{z}=0$ has a non-trivial solution, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{b^{2}}{q^{2}}=\frac{a c}{p r}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{c}^{2}}{\mathrm{r}^{2}}=\frac{\mathrm{ab}}{\mathrm{pq}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{a^{2}}{p^{2}}=\frac{b c}{q r}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{b^{2}}{q^{2}}=\frac{a c}{p r}$</span> </div>

answer:
<div class="solution">Since $\alpha_{1}, \alpha_{2}$ and $\beta_{1}, \beta_{2}$ are the roots of ax $^{2}$ $+b x+c=0$ and $p x^{2}+q x+r=0$ respectively
<br/> <br/>therefore
<br/> <br/>$$
<br/> <br/>\alpha_{1}+\alpha_{2}=\frac{-b}{a}, \alpha_{1} \alpha_{2}=\frac{c}{a}
<br/> <br/>$$
<br/> <br/>and $\beta_{1}+\beta_{2}=\frac{-q}{p}, \beta_{1} \beta_{2}=\frac{r}{p}$
<br/> <br/>Since the given system of equation has a non-trivial solution
<br/> <br/>$$
<br/> <br/>\therefore\left|\begin{array}{ll}
<br/> <br/>\alpha_{1} &amp; \alpha_{2} \\
<br/> <br/>\beta_{1} &amp; \beta_{2}
<br/> <br/>\end{array}\right|=0 \text { i.e. } \alpha_{1} \beta_{2}-\alpha_{2} \beta_{1}=0
<br/> <br/>$$
<br/> <br/>or $\quad \frac{\alpha_{1}}{\beta_{1}}=\frac{\alpha_{2}}{\beta_{2}}=\frac{\alpha_{1}+\alpha_{2}}{\beta_{1}+\beta_{2}}=\sqrt{\frac{\alpha_{1} \alpha_{2}}{\beta_{1} \beta_{2}}}$
<br/> <br/>$\Rightarrow \frac{\mathrm{pb}}{\mathrm{qa}}=\sqrt{\frac{\mathrm{pc}}{\mathrm{ra}}} \Rightarrow \frac{\mathrm{b}^{2}}{\mathrm{q}^{2}}=\frac{\mathrm{ac}}{\mathrm{pr}}$</div>


title: let-2-x-2-3-x-4-10-r-0-20-a-r-x-r-then-the-value-of-a-12-a-8-is
slug: let-2-x-2-3-x-4-10-r-0-20-a-r-x-r-then-the-value-of-a-12-a-8-is-93277
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mn>10</mn></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mn>20</mn></munderover><msub><mi>a</mi><mi>r</mi></msub><msup><mi>x</mi><mi>r</mi></msup></math>, then the value of $\frac{a_{8}}{a_{12}},$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span> </div>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mn>10</mn></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mn>20</mn></munderover><msub><mi>a</mi><mi>r</mi></msub><msup><mi>x</mi><mi>r</mi></msup></math></p><p>Replacing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mi>x</mi></mfrac></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mfrac><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mn>6</mn><mi>x</mi></mfrac><mo>+</mo><mn>4</mn></mrow></mfenced><mn>10</mn></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mn>20</mn></munderover><msub><mi>a</mi><mi>r</mi></msub><msup><mfenced><mfrac><mn>2</mn><mi>x</mi></mfrac></mfenced><mi>r</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mn>2</mn><mn>10</mn></msup><msup><mfenced><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mn>10</mn></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mn>20</mn></munderover><msub><mi>a</mi><mi>r</mi></msub><msup><mn>2</mn><mi>r</mi></msup><msup><mi>x</mi><mrow><mn>20</mn><mo>-</mo><mi>r</mi></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mn>2</mn><mn>10</mn></msup><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mn>20</mn></munderover><msub><mi>a</mi><mi>r</mi></msub><msup><mi>x</mi><mi>r</mi></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mn>20</mn></munderover><msub><mi>a</mi><mi>r</mi></msub><msup><mn>2</mn><mi>r</mi></msup><msup><mi>x</mi><mrow><mn>20</mn><mo>-</mo><mi>r</mi></mrow></msup></math></p><p>Comparing ccefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>8</mn></msup></math> both sides, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mn>10</mn></msup><msub><mi>a</mi><mn>8</mn></msub><mo>=</mo><msub><mi>a</mi><mn>12</mn></msub><msup><mn>2</mn><mn>12</mn></msup><mo>⇒</mo><mfrac><msub><mi>a</mi><mn>8</mn></msub><msub><mi>a</mi><mn>12</mn></msub></mfrac><mo>=</mo><mfrac><msup><mn>2</mn><mn>12</mn></msup><msup><mn>2</mn><mn>10</mn></msup></mfrac><mo>=</mo><mn>4</mn></math></p></div>


title: let-a-0-0-0-and-a-i-50-50-a-a-c-b-d-find-ab-c-ab-d-b-c-d-a-c-d
slug: let-a-0-0-0-and-a-i-50-50-a-a-c-b-d-find-ab-c-ab-d-b-c-d-a-c-d-18706
tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $A=\left[\begin{array}{ll}0 &amp; \alpha \\ 0 &amp; 0\end{array}\right]$ and $(A+I)^{50}-50 A=\left[\begin{array}{ll}a &amp; b \\ c &amp; d\end{array}\right]$,
<br/>find $a b c+a b d+b c d+a c d$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">As $\mathrm{A}^{2}=0, \mathrm{~A}^{k}=0 \forall k \geq 2$
<br/>Thus, $\quad(\mathrm{A}+\mathrm{I})^{50}=\mathrm{I}+50 \mathrm{~A} \Rightarrow(\mathrm{A}+\mathrm{I})^{50}-50 \mathrm{~A}=\mathrm{I}$
<br/>$\therefore \quad \mathrm{a}=1, \mathrm{~b}=0, \mathrm{c}=0, \mathrm{~d}=1$
<br/>$\mathrm{abc}+\mathrm{abd}+\mathrm{bed}+\mathrm{acd}=0$</div>


title: let-a-1-1-and-f-a-a-be-defined-as-f-x-x-x-for-all-x-a-then-f-x-is-1
slug: let-a-1-1-and-f-a-a-be-defined-as-f-x-x-x-for-all-x-a-then-f-x-is-1-82181
tags: ['Mathematics', 'Functions', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfenced close="]" open="["><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>A</mi></math> be defined as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mfenced close="|" open="|"><mi>x</mi></mfenced></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi>A</mi></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">many one and into function</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">one-many and into function</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">many-one and into function</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">one-one and onto function</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">one-one and onto function</span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mfenced close="|" open="|"><mi>x</mi></mfenced><mo>=</mo><mfenced close="" open="{"><mtable columnalign="center left"><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo></mtd><mtd><mi>x</mi><mo>≥</mo><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo></mtd><mtd><mi>x</mi><mo>&lt;</mo><mn>0</mn></mtd></mtr></mtable></mfenced></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/e6685d3b-a057-45ff-aab6-acf6a9903eba-d58.png" style="width: 329px; height: 250px;"/></p><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></math>, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>≤</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>≤</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Function is one-one and onto.</p></div>


title: let-a-1-2-1-1-1-1-1-3-1-and-10-b-4-5-1-2-0-2-2-3-if-b-is-the-inverse-of-a-then-the-value-of-is
tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
slug:
let-a-1-2-1-1-1-1-1-3-1-and-10-b-4-5-1-2-0-2-2-3-if-b-is-the-inverse-of-a-then-the-value-of-is-17896

question:
<div class="question">Let $A=\left[\begin{array}{ccc}1 &amp; -1 &amp; 1 \\ 2 &amp; 1 &amp; -3 \\ 1 &amp; 1 &amp; 1\end{array}\right]$ and $10 B=\left[\begin{array}{ccc}4 &amp; 2 &amp; 2 \\ -5 &amp; 0 &amp; \alpha \\ 1 &amp; -2 &amp; 3\end{array}\right]$
<br/>If $B$ is the inverse of $A$, then the value of $\alpha$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>4
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>3
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$-4$
<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"> <br/>5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>5</span> </div>

answer:
<div class="solution">$A=\left[\begin{array}{ccc}1 &amp; -1 &amp; 1 \\ 2 &amp; 1 &amp; -3 \\ 1 &amp; 1 &amp; 1\end{array}\right]$
<br/>$\therefore|A|=\left[\begin{array}{ccc}1 &amp; -1 &amp; 1 \\ 2 &amp; 1 &amp; -3 \\ 1 &amp; 1 &amp; 1\end{array}\right]$
<br/>$=1(1+3)+1(2+3)+1(2-1)=4+5+1=10$
<br/>$\Rightarrow \operatorname{Adj} A=\left[\begin{array}{ccc}4 &amp; -5 &amp; 1 \\ 2 &amp; 0 &amp; -2 \\ 2 &amp; 5 &amp; 3\end{array}\right]^T=\left[\begin{array}{ccc}4 &amp; 2 &amp; 2 \\ -5 &amp; 0 &amp; 5 \\ 1 &amp; -2 &amp; 3\end{array}\right]$
<br/>$\Rightarrow \quad B=A^{-1}=\frac{1}{10}\left[\begin{array}{ccc}4 &amp; 2 &amp; 2 \\ -5 &amp; 0 &amp; 5 \\ 1 &amp; -2 &amp; 3\end{array}\right]\left[\because A^{-1}=\frac{1}{|A|} \operatorname{Adj} A\right]$
<br/>$\Rightarrow 10 B=\left[\begin{array}{ccc}4 &amp; 2 &amp; 2 \\ -5 &amp; 0 &amp; 5 \\ 1 &amp; -2 &amp; 3\end{array}\right]$
<br/>Hence, $\alpha=5$</div>


title: let-a-1-2-3-4-5-and-r-be-a-relation-defined-by-r-x-y-x-y-a-x-y-5-then-r-is
slug: let-a-1-2-3-4-5-and-r-be-a-relation-defined-by-r-x-y-x-y-a-x-y-5-then-r-is-86410
tags: ['Mathematics', 'Sets and Relations', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> be a relation defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mo>{</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>:</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>A</mi><mo>,</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>5</mn><mo>}</mo></math>. Then, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">reflexive and symmetric but not transitive</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">an equivalence relation</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">neither reflexive nor transitive</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">neither reflexive nor symmetric but transitive</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">neither reflexive nor transitive</span> </div>

answer:
<div class="solution"><p>We first write the elements of the set <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>.</p><p>i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mo>{</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>,</mo><mo>(</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>,</mo><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>}</mo></math></p><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>∉</mo><mi>R</mi><mo>⇒</mo><mi>R</mi></math> is not reflexive. </p><p>Now as,<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>∈</mo><mi>R</mi><mo>⇒</mo><mo>(</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>∈</mo><mi>R</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>∈</mo><mi>R</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>∈</mo><mi>R</mi></math></p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is symmetric </p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>∈</mo><mi>R</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>∈</mo><mi>R</mi></math></p><p>Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is not transitive.</p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is symmetric but neither reflexive nor transitive.</p></div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
let-a-1-a-2-a-3-be-a-harmonic-progression-with-a-1-5-and-a-20-25-the-least-positive-integer-n-for-which-a-n-0-is

slug:
let-a-1-a-2-a-3-be-a-harmonic-progression-with-a-1-5-and-a-20-25-the-least-positive-integer-n-for-which-a-n-0-is-78944

question:
<div class="question">Let $a_1, a_2, a_3 \ldots$ be a harmonic progression with $a_1=5$ and $a_{20}=25$. The least positive integer $n$ for which $a_n &lt; 0$, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>22
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>23
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>24
<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"> <br/>25</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>25</span> </div>

answer:
<div class="solution">It is given that $a_1, a_2, a_3 \ldots$. are in HP. Therefore, $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldots \ldots .$. are in AP. Let $d$ be the common difference of the AP.
<br/>$\therefore \quad \frac{1}{a_n}=\frac{1}{a_1}+(n-1) d$ and $\frac{1}{a_{20}}=\frac{1}{a_1}+19 d$
<br/>$\Rightarrow \quad \frac{1}{a_n}=\frac{1}{5}+(n-1) d$ and $\frac{1}{25}=\frac{1}{5}+19 d$
<br/>$\Rightarrow \quad \frac{1}{a_n}=\frac{1}{5}+(n-1) d$ and $d=\frac{-4}{19 \times 25}$
<br/>$\Rightarrow \quad \frac{1}{a_n}=\frac{1}{5}-\frac{4(n-1)}{19 \times 25}$
<br/>$\Rightarrow \quad \frac{1}{a_n}=\frac{95-4 n+4}{19 \times 25}$
<br/>$\Rightarrow \quad a_n=\frac{99-4 n}{19 \times 25}$
<br/>Now, $a_n &lt; 0$
<br/>$\Rightarrow \quad \frac{99-4 n}{19 \times 25} &lt; 0$
<br/>$\Rightarrow \quad 99-4 n &lt; 0$
<br/>$\Rightarrow \quad 4 n&gt;99$
<br/>$\Rightarrow \quad n&gt;24 \frac{3}{4} \Rightarrow n \geq 25$</div>


title: let-a-1-a-2-a-3-be-terms-on-ap-if-a-1-a-2-a-q-a-1-a-2-a-p-q-2-p-2-p-q-then-a-21-a-6-equals
slug: let-a-1-a-2-a-3-be-terms-on-ap-if-a-1-a-2-a-q-a-1-a-2-a-p-q-2-p-2-p-q-then-a-21-a-6-equals-41272
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $a_{1}, a_{2}, a_{3} \ldots \ldots \ldots \ldots .$ be terms on A.P. If
<br/>$\frac{a_{1}+a_{2}+\ldots \ldots \ldots a_{p}}{a_{1}+a_{2}+\ldots \ldots \ldots .+a_{q}}=\frac{p^{2}}{q^{2}}, p \neq q,$ then $\frac{a_{6}}{a_{21}}$
<br/>equals</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{41}{11}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{7}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{7}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{11}{41}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{11}{41}$</span> </div>

answer:
<div class="solution">$\frac{\frac{p}{2}\left[2 a_{1}+(p-1) d\right]}{\frac{q}{2}\left[2 a_{1}+(q-1) d\right]}=\frac{p^{2}}{q^{2}}$
<br/>$\Rightarrow \frac{2 a_{1}+(p-1) d}{2 a_{1}+(q-1) d}=\frac{p}{q}$
<br/>$\frac{a_{1}+\left(\frac{p-1}{2}\right) d}{a_{1}+\left(\frac{q-1}{2}\right) d}=\frac{p}{q}$ For $\frac{a_{6}}{a_{21}}, \quad p=11, q=41$</div>


title: let-a-1-a-2-a-40-be-in-ap-and-h-1-h-2-h-10-be-in-hp-if-a-1-h-1-2-and-a-10-h-10-3-then-a-4-h-7-is
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
slug:
let-a-1-a-2-a-40-be-in-ap-and-h-1-h-2-h-10-be-in-hp-if-a-1-h-1-2-and-a-10-h-10-3-then-a-4-h-7-is-49115

question:
<div class="question">Let $a_1, a_2, \ldots \ldots a_{40}$ be in AP and $h_1, h_2, \ldots . h_{10}$ be in HP. If $a_1=h_1=2$ and $a_{10}=h_{10}=3$, then $a_4 h_7$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>2
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>3
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>5
<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"> <br/>6</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>6</span> </div>

answer:
<div class="solution">Let $d$ be the common difference of the AP. Then,
<br/>$\begin{aligned}
<br/>&amp; a_{10}=3 \Rightarrow a_1+9 d=3 \\
<br/>\Rightarrow &amp; 2+9 d=3 \Rightarrow d=\frac{1}{9} \\
<br/>\therefore &amp; a_4=a_1+3 d=2+\frac{1}{3}=\frac{7}{3}
<br/>\end{aligned}$
<br/>Let $D$ be the common difference of $\frac{1}{h_1}, \frac{1}{h_2}, \ldots .-\frac{1}{h_{10}}$.
<br/>Then, $\quad h_{10}=3$
<br/>$\begin{array}{ll}
<br/>\Rightarrow &amp; \frac{1}{h_{10}}=\frac{1}{3} \Rightarrow \frac{1}{2}+9 D=\frac{1}{3} \\
<br/>\Rightarrow &amp; 9 D=-\frac{1}{6} \Rightarrow D=-\frac{1}{54} \\
<br/>\therefore &amp; \frac{1}{h_7}=\frac{1}{h_1}+6 D=\frac{1}{2}-\frac{1}{9}=\frac{7}{18} \\
<br/>\Rightarrow &amp; h_7=\frac{18}{7} \\
<br/>\therefore &amp; a_4 h_7=\frac{7}{3} \times \frac{18}{7}=6
<br/>\end{array}$</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
let-a-and-b-are-two-independent-events-if-the-probability-that-both-a-and-b-occur-together-is-1-6-and-the-probability-that-neither-of-them-occurs-is-1

slug:
let-a-and-b-are-two-independent-events-if-the-probability-that-both-a-and-b-occur-together-is-1-6-and-the-probability-that-neither-of-them-occurs-is-1-14310

question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> are two independent events. If the probability that both <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> occur together is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>6</mn></mfrac></math> and the probability that neither of them occurs is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math>, then the probability of occurrence of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mtext> or </mtext><mn>1</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext> or </mtext><mfrac><mn>1</mn><mn>3</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext> or </mtext><mfrac><mn>1</mn><mn>4</mn></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mtext> or </mtext><mfrac><mn>1</mn><mn>4</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext> or </mtext><mfrac><mn>1</mn><mn>3</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>A</mi><mo>∩</mo><mi>B</mi></mrow></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><msup><mi>A</mi><mi>C</mi></msup><mo>∩</mo><msup><mi>B</mi><mi>C</mi></msup></mrow></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><msup><mfenced><mrow><mi>A</mi><mo>∪</mo><mi>B</mi></mrow></mfenced><mi>C</mi></msup><mo>=</mo><mi>P</mi><mfenced><mrow><msup><mi>A</mi><mi>C</mi></msup><mo>∩</mo><msup><mi>B</mi><mi>C</mi></msup></mrow></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>1</mn><mo>-</mo><mi>P</mi><mfenced><mrow><mi>A</mi><mo>∪</mo><mi>B</mi></mrow></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>P</mi><mfenced><mrow><mi>A</mi><mo>∪</mo><mi>B</mi></mrow></mfenced><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math></p><p>But <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>A</mi><mo>∪</mo><mi>B</mi></mrow></mfenced><mo>=</mo><mi>P</mi><mfenced><mi>A</mi></mfenced><mo>+</mo><mi>P</mi><mfenced><mi>B</mi></mfenced><mo>-</mo><mi>P</mi><mfenced><mrow><mi>A</mi><mo>∩</mo><mi>B</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>P</mi><mfenced><mi>A</mi></mfenced><mo>+</mo><mi>P</mi><mfenced><mi>B</mi></mfenced><mo>=</mo><mfrac><mn>5</mn><mn>6</mn></mfrac><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo><mo> </mo><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> are independent events</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>P</mi><mfenced><mrow><mi>A</mi><mo>∩</mo><mi>B</mi></mrow></mfenced><mo>=</mo><mi>P</mi><mfenced><mi>A</mi></mfenced><mi>P</mi><mfenced><mi>B</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>P</mi><mfenced><mi>A</mi></mfenced><mi>P</mi><mfenced><mi>B</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced close="]" open="["><mrow><mi>P</mi><mfenced><mi>A</mi></mfenced><mo>-</mo><mi>P</mi><mfenced><mi>B</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced close="]" open="["><mrow><mi>P</mi><mfenced><mi>A</mi></mfenced><mo>+</mo><mi>P</mi><mfenced><mi>B</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>P</mi><mfenced><mi>A</mi></mfenced><mi>P</mi><mfenced><mi>B</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>25</mn><mn>36</mn></mfrac><mo>-</mo><mfrac><mn>4</mn><mn>6</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>36</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>P</mi><mfenced><mi>A</mi></mfenced><mo>-</mo><mi>P</mi><mfenced><mi>B</mi></mfenced><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced></math></p><p>On solving Eqs. (i) and (ii), we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mi>A</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext> or </mtext><mfrac><mn>1</mn><mn>3</mn></mfrac></math></p><p>Hence, option (b) is correct.</p></div>


title: let-a-and-b-are-two-noncollinear-unit-vectors-if-u-a-a-b-b-and-v-a-b-then-v-is-equal-to
slug: let-a-and-b-are-two-noncollinear-unit-vectors-if-u-a-a-b-b-and-v-a-b-then-v-is-equal-to-46199
tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="italic">a</mi><mo mathvariant="italic">^</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="italic">b</mi><mo mathvariant="italic">^</mo></mover></math> are two non-collinear unit vectors. If <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>u</mi><mo>→</mo></mover><mo>=</mo><mover><mi>a</mi><mo>^</mo></mover><mo>-</mo><mfenced><mrow><mover><mi>a</mi><mo mathvariant="italic">^</mo></mover><mo mathvariant="italic">·</mo><mover><mi>b</mi><mo mathvariant="italic">^</mo></mover></mrow></mfenced><mover><mi>b</mi><mo>^</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>v</mi><mo>→</mo></mover><mo>=</mo><mover><mi mathvariant="italic">a</mi><mo mathvariant="italic">^</mo></mover><mo mathvariant="italic">×</mo><mover><mi mathvariant="italic">b</mi><mo mathvariant="italic">^</mo></mover></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mover><mi>v</mi><mo>→</mo></mover><mo>|</mo></math> is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mover><mi>u</mi><mo>→</mo></mover></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mover><mi>u</mi><mo>→</mo></mover></mfenced><mo>+</mo><mfenced close="|" open="|"><mrow><mover><mi>v</mi><mo>→</mo></mover><mo>.</mo><mover><mi>a</mi><mo>^</mo></mover></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced close="|" open="|"><mover><mi>v</mi><mo>→</mo></mover></mfenced></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mover><mi>v</mi><mo>→</mo></mover></mfenced><mo>+</mo><mover><mi>u</mi><mo>→</mo></mover><mo>·</mo><mfenced><mrow><mover><mi mathvariant="italic">a</mi><mo mathvariant="italic">^</mo></mover><mo>+</mo><mover><mi mathvariant="italic">b</mi><mo mathvariant="italic">^</mo></mover></mrow></mfenced></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mover><mi>u</mi><mo>→</mo></mover></mfenced></math></span> </div>

answer:
<div class="solution"><p>We have, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mover><mi>a</mi><mo>^</mo></mover></mfenced><mo>=</mo><mfenced close="|" open="|"><mover><mi>b</mi><mo>^</mo></mover></mfenced><mo>=</mo><mn>1</mn></math></p><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>v</mi><mo>→</mo></mover><mo>=</mo><mover><mi>a</mi><mo>^</mo></mover><mo>×</mo><mover><mi mathvariant="italic">b</mi><mo mathvariant="italic">^</mo></mover><mo>=</mo><mfenced close="|" open="|"><mover><mi mathvariant="italic">a</mi><mo mathvariant="italic">^</mo></mover></mfenced><mfenced close="|" open="|"><mover><mi>b</mi><mo>^</mo></mover></mfenced><mi>sin</mi><mi>θ</mi><mover><mi mathvariant="italic">n</mi><mo mathvariant="italic">^</mo></mover><mo>=</mo><mi>sin</mi><mi>θ</mi><mover><mi>n</mi><mo>^</mo></mover></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> is the angle between <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>^</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>b</mi><mo>^</mo></mover></math>. </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo> </mo><mo> </mo><mfenced close="|" open="|"><mover><mi>v</mi><mo>→</mo></mover></mfenced><mo>=</mo><mi>sin</mi><mi>θ</mi></math></p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>u</mi><mo>→</mo></mover><mo>=</mo><mover><mi>a</mi><mo>^</mo></mover><mo>-</mo><mfenced><mrow><mover><mi mathvariant="italic">a</mi><mo mathvariant="italic">^</mo></mover><mo mathvariant="italic">·</mo><mover><mi mathvariant="italic">b</mi><mo mathvariant="italic">^</mo></mover></mrow></mfenced><mover><mi>b</mi><mo>^</mo></mover><mo>=</mo><mover><mi>a</mi><mo>^</mo></mover><mo>-</mo><mfenced><mrow><mfenced close="|" open="|"><mover><mi>a</mi><mo>^</mo></mover></mfenced><mfenced close="|" open="|"><mover><mi>b</mi><mo>^</mo></mover></mfenced><mi>cos</mi><mi>θ</mi></mrow></mfenced><mover><mi>b</mi><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>u</mi><mo>→</mo></mover><mo>=</mo><mover><mi>a</mi><mo mathvariant="italic">^</mo></mover><mo>-</mo><mfenced><mrow><mi>cos</mi><mi>θ</mi></mrow></mfenced><mover><mi>b</mi><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfenced close="|" open="|"><mover><mi>u</mi><mo>→</mo></mover></mfenced><mo>=</mo><msqrt><msup><mfenced close="|" open="|"><mover><mi>a</mi><mo>^</mo></mover></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced close="|" open="|"><mrow><mo>-</mo><mi>cos</mi><mi>θ</mi><mfenced close="|" open="|"><mover><mi>b</mi><mo>^</mo></mover></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced close="|" open="|"><mover><mi>a</mi><mo>^</mo></mover></mfenced><mfenced close="|" open="|"><mrow><mo>-</mo><mi>cos</mi><mi>θ</mi><mover><mi>b</mi><mo>^</mo></mover></mrow></mfenced><mi>cos</mi><mi>θ</mi></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfenced close="|" open="|"><mover><mi>u</mi><mo>→</mo></mover></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>θ</mi><mo>-</mo><mn>2</mn><msup><mi>cos</mi><mn>2</mn></msup><mi>θ</mi></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced close="|" open="|"><mover><mi>u</mi><mo>→</mo></mover></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>θ</mi></msqrt><mo>=</mo><msqrt><msup><mi>sin</mi><mn>2</mn></msup><mi>θ</mi></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfenced close="|" open="|"><mover><mi>u</mi><mo>→</mo></mover></mfenced><mo>=</mo><mi>sin</mi><mi>θ</mi></math></p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mover><mi>v</mi><mo>→</mo></mover></mfenced><mo>=</mo><mfenced close="|" open="|"><mover><mi>u</mi><mo>→</mo></mover></mfenced></math></p></div>


title: let-a-and-b-be-two-sets-such-that-a-x-b-x-and-a-x-b-x-for-same-set-x-then
slug: let-a-and-b-be-two-sets-such-that-a-x-b-x-and-a-x-b-x-for-same-set-x-then-96708
tags: ['Mathematics', 'Sets and Relations', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> be two sets such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∩</mo><mi>X</mi><mo>=</mo><mi>B</mi><mo>∩</mo><mi>X</mi><mo>=</mo><mi>ϕ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∪</mo><mi>X</mi><mo>=</mo><mi>B</mi><mo>∪</mo><mi>X</mi></math> for same set <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>. Then,</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>B</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>X</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mi>X</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∪</mo><mi>B</mi><mo>=</mo><mi>X</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>B</mi></math></span> </div>

answer:
<div class="solution"><p>Given, that, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∪</mo><mi>X</mi><mo>=</mo><mi>B</mi><mo>∪</mo><mi>X</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>A</mi><mo>∩</mo><mo>(</mo><mi>A</mi><mo>∪</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>A</mi><mo>∩</mo><mo>(</mo><mi>B</mi><mo>∪</mo><mi>X</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>(</mo><mi>A</mi><mo>∩</mo><mi>A</mi><mo>)</mo><mo>∪</mo><mo>(</mo><mi>A</mi><mo>∩</mo><mi>X</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>)</mo><mo>∪</mo><mo>(</mo><mi>A</mi><mo>∩</mo><mi>X</mi><mo>)</mo></math></p><p>[Using distributive law]</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>A</mi><mo>∪</mo><mi>ϕ</mi><mo>=</mo><mo>(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>)</mo><mo>∪</mo><mi>ϕ</mi><mo> </mo><mo>[</mo><mo>∵</mo><mi>A</mi><mo>∩</mo><mi>X</mi><mo>=</mo><mi>ϕ</mi><mo>]</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>A</mi><mo>=</mo><mi>A</mi><mo>∩</mo><mi>B</mi></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math></p><p>Again, consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∪</mo><mi>X</mi><mo>=</mo><mi>B</mi><mo>∪</mo><mi>X</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>B</mi><mo>∩</mo><mo>(</mo><mi>A</mi><mo>∪</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>B</mi><mo>∩</mo><mo>(</mo><mi>B</mi><mo>∪</mo><mi>X</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>(</mo><mi>B</mi><mo>∩</mo><mi>A</mi><mo>)</mo><mo>∪</mo><mo>(</mo><mi>B</mi><mo>∩</mo><mi>X</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>B</mi><mo>∩</mo><mi>B</mi><mo>)</mo><mo>∪</mo><mo>(</mo><mi>B</mi><mo>∩</mo><mi>X</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>(</mo><mi>B</mi><mo>∩</mo><mi>A</mi><mo>)</mo><mo>∪</mo><mi>ϕ</mi><mo>=</mo><mi>B</mi><mo>∪</mo><mi>ϕ</mi><mo>[</mo><mi>B</mi><mo>∩</mo><mi>X</mi><mo>=</mo><mi>ϕ</mi><mo>]</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>=</mo><mi>B</mi></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>ii</mi><mo>)</mo></math></p><p>Thus from Eq. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>ii</mi><mo>)</mo></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>B</mi></math></p></div>


title: let-a-and-b-be-two-sets-then-a-b-a-b-is-equal-to-1
slug: let-a-and-b-be-two-sets-then-a-b-a-b-is-equal-to-1-60431
tags: ['Mathematics', 'Sets and Relations', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let A and B be two sets then $(\mathrm{A} \cup \mathrm{B}) \cup\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{A}^{\prime}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{A}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">B'</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{A}^{\prime}$</span> </div>

answer:
<div class="solution">$\text { From Venn-Euler's Diagram. }$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/8wxGXrdi91NGzVF0HoaaVQ_kkP69EunyYFwoLp4RYVo.original.fullsize.png"/><br/> <br/>$\therefore(\mathrm{A} \cup \mathrm{B})^{\prime} \cup\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)=\mathrm{A}^{\prime}$</div>


title: let-a-b-and-c-be-three-vectors-satisfying-a-b-a-c-a-c-1-b-4-and-b-c-15-if-b-2-c-a-then-equals
slug: let-a-b-and-c-be-three-vectors-satisfying-a-b-a-c-a-c-1-b-4-and-b-c-15-if-b-2-c-a-then-equals-35258
tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-4</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $a, b$ and $c$ be three vectors satisfying $a \times b=$ $(a \times c),|a|=|c|=1,|b|=4$ and $|b \times c|=\sqrt{15}$. If $b-$
<br/> <br/>$2 \mathrm{c}=\lambda \mathrm{a},$ then $\lambda$ equals</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">-4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}$. Given, $|\mathrm{b} \times \mathrm{c}|=\sqrt{15}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow|\mathrm{b} \| \mathrm{c}| \sin \theta=\sqrt{15} \\
<br/> <br/>\Rightarrow \sin \theta=\frac{\sqrt{15}}{|\mathrm{~b}| \mathrm{c} \mid} \\
<br/> <br/>\Rightarrow \sin \theta=\sqrt{\frac{15}{4 \times 1}}=\frac{\sqrt{15}}{4} \\
<br/> <br/>\quad \therefore \cos \theta=\sqrt{1-\frac{15}{16}}=\frac{1}{\sqrt{16}}=\frac{1}{4}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Now given, $\mathrm{b}-2 \mathrm{c}=\lambda \mathrm{a} \Rightarrow\left|\mathrm{b}-2 \mathrm{c}^{2}\right|=\mid \lambda \mathrm{a}^{2}$
<br/> <br/>$$
<br/> <br/>\Rightarrow|\mathrm{b}|^{2}+4|\mathrm{c}|-4(\mathrm{~b} \cdot \mathrm{c})=\lambda^{2}|\mathrm{a}|^{2}
<br/> <br/>$$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow 16+4-4|\mathrm{~b} \| \mathrm{c}| \cos \theta=\lambda^{2} \\
<br/> <br/>\Rightarrow 20-16 \cos \theta=\lambda^{2} \\
<br/> <br/>\Rightarrow 20-4=\lambda^{2} \Rightarrow \lambda^{2}=16\left(\therefore \cos \theta=\frac{1}{4}\right) \\
<br/> <br/>\Rightarrow \lambda=\pm 4
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
let-a-b-be-the-solutions-of-x-2-p-x-1-0-and-c-d-be-the-solution-of-x-2-q-x-1-0-if-a-c-b-c-and-a-d-b-d-are-the-solution-of-x-2-a-x-0-then-is-equal-to

slug:
let-a-b-be-the-solutions-of-x-2-p-x-1-0-and-c-d-be-the-solution-of-x-2-q-x-1-0-if-a-c-b-c-and-a-d-b-d-are-the-solution-of-x-2-a-x-0-then-is-equal-to-68818

question:
<div class="question">Let $a, b$ be the solutions of $x^2+p x+1=0$ and $c, d$ be the solution of $x^2+q x+1=0$. If $(a-c)(b-c)$ and $(a+d)(b+d)$ are the solution of $x^2+a x+\beta=0$, then $\beta$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$p+q$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$p-q$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$p^2+q^2$
<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"> <br/>$q^2-p^2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$q^2-p^2$</span> </div>

answer:
<div class="solution">Since, $a+b=-p, a b=1$ .....(i)
<br/>and $c+d=-q, c d=1$
<br/>Now $(a-c)(b-c)$ and $(a+d)(b+d)$ are the roots of $x^2+a x+\beta=0$
<br/>$$
<br/>\begin{aligned}
<br/>&amp; (a-c)(b-c)(a+d)(b+d)=\beta \\
<br/>&amp; \Rightarrow\left(a b-a c-b c+c^2\right)\left(a b+a d+b d+d^2\right)=\beta \\
<br/>&amp; \Rightarrow\left\{1-c(a+b)+c^2\right\}\left\{1+d(a+b)+d^2\right\}=\beta \\
<br/>&amp; \Rightarrow \quad\left(1+p c+c^2\right)\left(1-p d+d^2\right)=\beta \\
<br/>&amp; \Rightarrow 1-p d+d^2+p c-p^2 c d+p c d^2+c^2-p c^2 d \\
<br/>&amp; +c^2 d^2=\beta \\
<br/>&amp; \Rightarrow 1-p d+d^2+p c-p^2+p d+c^2-p c+1=\beta \\
<br/>&amp; {[\because c d=1]} \\
<br/>&amp; \Rightarrow \quad 2+d^2+c^2-p^2=\beta \\
<br/>&amp;
<br/>\end{aligned}
<br/>$$
<br/>$\begin{aligned} \Rightarrow \quad 2 c d+c^2+d^2-p^2 &amp; =\beta \\ (c+d)^2-p^2 &amp; =\beta \\ q^2-p^2 &amp; =\beta \quad[\because(c+d)=-q]\end{aligned}$</div>


tags: ['Mathematics', 'Properties of Triangles', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
let-a-b-c-be-an-acuteangled-triangle-and-let-d-be-the-midpoint-of-b-c-if-a-b-a-d-then-tan-b-tan-c-equal

slug:
let-a-b-c-be-an-acuteangled-triangle-and-let-d-be-the-midpoint-of-b-c-if-a-b-a-d-then-tan-b-tan-c-equal-75251

question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mi>C</mi></math> be an acute-angled triangle and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math> be the mid-point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi><mi>D</mi></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>tan</mi><mi>B</mi></mrow><mrow><mi>tan</mi><mi>C</mi></mrow></mfrac></math> equal</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>3</mn></msqrt></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </div>

answer:
<div class="solution"><p>In  <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo><mi>A</mi><mi>B</mi><mi>C</mi><mo>,</mo><mi>D</mi></math> be the mid-point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi><mi>D</mi><mo>⇒</mo><mi mathvariant="normal">Δ</mi><mi>A</mi><mi>B</mi><mi>D</mi></math> is isosceles.</p><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>M</mi></math> be the perpendicular drawn from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>A</mi><mi>M</mi></math> is perpendicular bisector of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>D</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>M</mi><mo>=</mo><mi>M</mi><mi>D</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi mathvariant="normal">Δ</mi><mi>A</mi><mi>B</mi><mi>M</mi><mo>,</mo><mo> </mo><mi>tan</mi><mi>B</mi><mo>=</mo><mfrac><mrow><mi>A</mi><mi>M</mi></mrow><mrow><mi>B</mi><mi>M</mi></mrow></mfrac></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/3f893ec8-7600-4ae1-9129-15ccf51bb877-d36.png" style="width: 375px; height: 170px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>Δ</mi><mi>A</mi><mi>M</mi><mi>C</mi><mo>,</mo><mo> </mo><mi>tan</mi><mi>C</mi><mo>=</mo><mfrac><mrow><mi>A</mi><mi>M</mi></mrow><mrow><mi>C</mi><mi>M</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfrac><mrow><mi>tan</mi><mi>B</mi></mrow><mrow><mi>tan</mi><mi>C</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>C</mi><mi>M</mi></mrow><mrow><mi>B</mi><mi>M</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mi>B</mi><mi>M</mi></mrow><mrow><mi>B</mi><mi>M</mi></mrow></mfrac><mo>=</mo><mn>3</mn><mo>  </mo><mfenced close="]" open="["><mrow><mo>∵</mo><mi>C</mi><mi>M</mi><mo>=</mo><mi>D</mi><mi>M</mi><mo>+</mo><mi>C</mi><mi>D</mi></mrow></mfenced></math></p></div>


tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
let-a-b-c-r-and-the-system-of-equations-1-a-x-y-z-0-x-1-b-y-z-0-and-x-y-1-c-z-0-has-infinitely-many-solutions-the-minimum-value-of-a-b-c-is

slug:
let-a-b-c-r-and-the-system-of-equations-1-a-x-y-z-0-x-1-b-y-z-0-and-x-y-1-c-z-0-has-infinitely-many-solutions-the-minimum-value-of-a-b-c-is-78571

question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><msup><mi>R</mi><mo>+</mo></msup></math> and the system of equations <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>-</mo><mi>a</mi><mo>)</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>+</mo><mo>(</mo><mn>1</mn><mo>-</mo><mi>b</mi><mo>)</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mo>(</mo><mn>1</mn><mo>-</mo><mi>c</mi><mo>)</mo><mi>z</mi><mo>=</mo><mn>0</mn></math> has infinitely many solutions, the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mi>c</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msqrt><mn>3</mn></msqrt></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi>b</mi></mrow></mfenced><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi>c</mi></mrow></mfenced><mi>z</mi><mo>=</mo><mn>0</mn></math></p><p>has many solutions </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfenced close="|" open="|"><mtable><mtr><mtd><mn>1</mn><mo>-</mo><mi>a</mi></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn><mo>-</mo><mi>b</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn><mo>-</mo><mi>c</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>a</mi><mi>b</mi><mo>+</mo><mi>a</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi><mi>c</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfrac><mrow><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>c</mi></mrow><mn>3</mn></mfrac><mo>≥</mo><msup><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><msup><mi>c</mi><mn>2</mn></msup></mrow></mfenced><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mspace linebreak="newline"></mspace><mo>⇒</mo><msup><mfenced><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mfenced><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>≥</mo><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Minimum of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mi>c</mi><mo>=</mo><msup><mfenced><mn>3</mn></mfenced><mn>3</mn></msup><mo>=</mo><mn>27</mn></math></p></div>


tags: ['Mathematics', 'Heights and Distances', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
let-be-the-solution-of-1-6-s-i-n-2-1-6-c-o-s-2-10-in-0-4-if-the-shadow-of-a-vertical-pole-is-3-1-of-its-height-then-the-altitude-of-the-sun-is

slug:
let-be-the-solution-of-1-6-s-i-n-2-1-6-c-o-s-2-10-in-0-4-if-the-shadow-of-a-vertical-pole-is-3-1-of-its-height-then-the-altitude-of-the-sun-is-97933

question:
<div class="question">Let $\alpha$ be the solution of $16^{\sin ^2 \theta}+16^{\cos ^2 \theta}=10$ in $\left(0, \frac{\pi}{4}\right)$. If the shadow of a vertical pole is $\frac{1}{\sqrt{3}}$ of its height, then the altitude of the sun is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$\alpha$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$\frac{\alpha}{2}$
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$2 \alpha$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$\frac{\alpha}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$2 \alpha$
<br/></span> </div>

answer:
<div class="solution">We have,
<br/>$16^{\sin ^2 \theta}+16^{\cos ^2 \theta}=10$
<br/>$\Rightarrow \quad 16^{\sin ^2 \theta}+16^{1-\sin ^2 \theta}=10$
<br/>$\Rightarrow x+\frac{16}{x}=10$, where $x=16^{\sin ^2 \theta}$
<br/>$\Rightarrow \quad x^2-10 x+16=0 \Rightarrow x=2,8$
<br/>$\therefore \quad 16^{\sin ^2 \theta}=2,8 \Rightarrow 2^{4 \sin ^2 \theta}=2,2^3$
<br/>$\Rightarrow \quad 4 \sin ^2 \theta=1,3 \Rightarrow \sin ^2 \theta=\frac{1}{4},\left(\frac{\sqrt{3}}{2}\right)^2$
<br/>$\Rightarrow \quad \sin \theta=\frac{1}{2}, \frac{\sqrt{3}}{2} \Rightarrow \theta=\frac{\pi}{6}, \frac{\pi}{3}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/S5pOq2wkF-enUh0phx2RJpTLusnmaGSnw_vsT1ydx70.original.fullsize.png"/><br/> <br/>Let the altitude of the sun be $\theta$. Then,
<br/>$$
<br/>\begin{aligned}
<br/>&amp; &amp; \tan \theta &amp; =\frac{h}{h / \sqrt{3}} \Rightarrow \tan \theta=\sqrt{3} \\
<br/>\Rightarrow \quad &amp; &amp; \theta &amp; =\pi / 3 \Rightarrow \theta=2 \alpha
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Nuclear Physics', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
title:
let-binding-energy-per-nucleon-of-nucleus-is-denoted-by-e-bn-and-radius-of-nucleus-is-denoted-as-r-if-mass-number-of-nuclei-a-b-and-64-and-125-respect

slug:
let-binding-energy-per-nucleon-of-nucleus-is-denoted-by-e-bn-and-radius-of-nucleus-is-denoted-as-r-if-mass-number-of-nuclei-a-b-and-64-and-125-respect-48911

question:
<div class="question">Let binding energy per nucleon of nucleus is denoted by $\mathrm{E}_{\mathrm{bn}}$ and radius of nucleus is denoted as $r$. If mass number of nuclei $\mathrm{A}, \mathrm{B}$ and 64 and 125 respectively then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(r_{A} &lt; r_{B}, E_{b n A} &lt; E_{b n B}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(r_{A} &lt; r_{B}, E_{b n A} &lt; E_{b n A}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(r_{A}=\frac{4}{5} r_{B}, E_{b n A} &lt; E_{b n B}\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(r_{A} &lt; r_{B}, E_{b n A} &gt; E_{b n B}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(r_{A} &lt; r_{B}, E_{b n A} &gt; E_{b n B}\)</span> </div>

answer:
<div class="solution">$\mathrm{r}=\mathrm{r}_{0}(\mathrm{~A})^{1 / 3}$
<br/> <br/>r increases with increasing A mass number So, $r_{A}<r_ as="" is="" mass="" number="" of="" smaller=""></r_> <br/>$\mathrm{E}_{\mathrm{bn}}$ decreases with increasing $\mathrm{A}$ for $\mathrm{A}&gt;56$ ${ }^{56} \mathrm{Fe}$ has highest $\mathrm{E}_{\mathrm{bn}}$ value. So, $E_{b n}$ for $A=64$ is larger as compared to $\mathrm{E}_{\mathrm{bn}}$ for nucleus with $\mathrm{A}=125$
<br/> <br/>$\mathrm{E}_{\mathrm{bnA}}&gt;\mathrm{E}_{\mathrm{bnB}}$</div>


title: let-cos-4-5-and-let-sin-5-13-where-0-4-then-tan-2
slug: let-cos-4-5-and-let-sin-5-13-where-0-4-then-tan-2-52676
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced><mo>=</mo><mfrac><mn>4</mn><mn>5</mn></mfrac></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mo>=</mo><mfrac><mn>5</mn><mn>13</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>≤</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mn>2</mn><mi>α</mi><mo>=</mo></math></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>20</mn><mn>7</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>25</mn><mn>16</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>56</mn><mn>33</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>19</mn><mn>2</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>56</mn><mn>33</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>≤</mo><mfrac><mi>π</mi><mn>4</mn></mfrac><mo>;</mo><mo> </mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>≤</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo>(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>)</mo><mo>=</mo><mn>4</mn><mo>/</mo><mn>5</mn><mo>⇒</mo><mi>tan</mi><mo>(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>)</mo><mo>=</mo><mn>3</mn><mo>/</mo><mn>4</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo>(</mo><mi>α</mi><mo>-</mo><mi>β</mi><mo>)</mo><mo>=</mo><mn>5</mn><mo>/</mo><mn>13</mn><mo>⇒</mo><mi>α</mi><mo>-</mo><mi>β</mi><mo>∈</mo><msub><mi>θ</mi><mn>1</mn></msub></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo>(</mo><mi>α</mi><mo>-</mo><mi>β</mi><mo>)</mo><mo>=</mo><mn>5</mn><mo>/</mo><mn>12</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mn>2</mn><mi>α</mi><mo>=</mo><mi>tan</mi><mo>(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>α</mi><mo>-</mo><mi>β</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>tan</mi><mo>(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>)</mo><mo>+</mo><mi>tan</mi><mo>(</mo><mi>α</mi><mo>-</mo><mi>β</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>-</mo><mi>tan</mi><mo>(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>)</mo><mi>tan</mi><mo>(</mo><mi>α</mi><mo>-</mo><mi>β</mi><mo>)</mo></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>3</mn><mo>/</mo><mn>4</mn><mo>+</mo><mn>5</mn><mo>/</mo><mn>12</mn></mrow><mrow><mn>1</mn><mo>-</mo><mn>3</mn><mo>/</mo><mn>4</mn><mo>×</mo><mfrac><mn>5</mn><mn>12</mn></mfrac></mrow></mfrac><mo>=</mo><mfrac><mn>56</mn><mn>33</mn></mfrac></math></p></div>


title: let-e1234-and-f12-then-the-number-of-onto-functions-from-mathrme-to-mathrmf-is
slug: let-e1234-and-f12-then-the-number-of-onto-functions-from-mathrme-to-mathrmf-is-53710
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">14</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let \(E=\{1,2,3,4\}\) and \(F=\{1,2\}\). Then the number of onto functions from \(\mathrm{E}\) to \(\mathrm{F}\) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">14</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">16</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">If set \(A\) has \(m\) elements and set \(B\) has \(\mathrm{n}\) elements then number of onto functions from \(A\) to \(B\) is<br/>\(\sum_{r=1}^n(-1)^{n-r}{ }^n C_r r^m \text { where } 1 \leq n \leq m\)<br/>Here \(E=\{1,2,3,4\}, F=\{1,2\}\)<br/>\(m=4, n=2\)<br/>\(\therefore\) No. of onto functions from \(\mathrm{E}\) to \(\mathrm{F}\)<br/>\(\begin{aligned}<br/>&amp; =\sum_{r=1}^2(-1)^{2-r ~2} C_r(r)^4 \\<br/>&amp; =(-1)^2 C_1+{ }^2 C_2(2)^4=-2+16=14<br/>\end{aligned}\)</div>


title: let-f-be-the-function-defined-by-f-x-x-2-2-x-1-1-x-2-1-12-x-1-x-1
slug: let-f-be-the-function-defined-by-f-x-x-2-2-x-1-1-x-2-1-12-x-1-x-1-81315
tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{f}$ be the function defined by $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-1}{x^{2}-2|x-1|-1}, &amp; x \neq 1 \\ 1 / 2, &amp; x=1\end{array}\right.$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The function is continuous for all values of x</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The function is continuous only for $x&gt;1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The function is continuous at $\mathrm{x}=1$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">The function is not continuous at $x=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The function is not continuous at $x=1$</span> </div>

answer:
<div class="solution">For $x&lt;1, f(x)=\frac{x^{2}-1}{x^{2}+2 x-3}=\frac{x+1}{x+3}$
<br/> <br/>$\therefore \lim _{x \rightarrow 1^{-}} \mathrm{f}(\mathrm{x})=\frac{1}{2}$
<br/> <br/>For $x&gt;1, f(x)=\frac{x^{2}-1}{x^{2}-2 x+1}=\frac{x+1}{x-1}$
<br/> <br/>$\therefore \lim _{x \rightarrow 1^{+}} f(x)=\infty$
<br/> <br/>$\therefore$ The function is not continuous at $\mathrm{x}=1$.</div>


title: let-f-r-r-be-a-function-defined-by-f-x-x-n-x-m-where-m-n-then
slug: let-f-r-r-be-a-function-defined-by-f-x-x-n-x-m-where-m-n-then-41872
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a function defined by
<br/> <br/>$f(x)=\frac{x-m}{x-n},$ where $m \neq n,$ then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{f}$ is one-one onto</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{f}$ is one-one into</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{f}$ is many-one onto</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">fis many-one into</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}$ is one-one into</span> </div>

answer:
<div class="solution">Let $f: R \rightarrow R$ be a function defined by
<br/> <br/>$f(x)=\frac{x-m}{x-n}$
<br/> <br/>For any $(\mathrm{x}, \mathrm{y}) \in \mathrm{R}$
<br/> <br/>Let $\mathrm{f}(\mathrm{x})=\mathrm{f}(\mathrm{y})$
<br/> <br/>$\Rightarrow \frac{x-m}{x-n}=\frac{y-m}{y-n} \Rightarrow x=y$
<br/> <br/>$\therefore \mathrm{f}$ is one $-$ one
<br/> <br/>Let $\alpha \in \mathrm{R}$ such that $\mathrm{f}(\mathrm{x})=\alpha$
<br/> <br/>$$
<br/> <br/>\Rightarrow \alpha=\frac{x-m}{x-n} \Rightarrow(x-n) \alpha=x-m
<br/> <br/>$$
<br/> <br/>$\Rightarrow x \alpha-n \alpha=x-m$
<br/> <br/>$$
<br/> <br/>\Rightarrow x \alpha-x=n \alpha-m
<br/> <br/>$$
<br/> <br/>$\Rightarrow x(\alpha-1)=n \alpha-m$
<br/> <br/>$\Rightarrow \mathrm{x}=\frac{\mathrm{n} \alpha-\mathrm{m}}{\alpha-1} \cdot$ for $\alpha=1, \mathrm{x} \notin \mathrm{R}$
<br/> <br/>So, $\mathrm{f}$ is not onto.</div>


tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
let-f-r-r-be-a-function-satisfying-f-x-y-f-x-2-y-2-k-x-y-for-all-x-y-r-if-f-1-2-and-f-2-8-then-f-x-is-equal-to

slug:
let-f-r-r-be-a-function-satisfying-f-x-y-f-x-2-y-2-k-x-y-for-all-x-y-r-if-f-1-2-and-f-2-8-then-f-x-is-equal-to-94392

question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math> be a function satisfying <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi><mi>x</mi><mi>y</mi></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>R</mi></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>2</mn></mfenced><mo>=</mo><mn>8</mn></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mi>x</mi><mo>-</mo><mn>4</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>2</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mi>x</mi><mo>-</mo><mn>6</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math></span> </div>

answer:
<div class="solution"><p>We have, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi><mi>x</mi><mi>y</mi></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>R</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mi>f</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>-</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow><mi>y</mi></mfrac><mo>=</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>k</mi><mi>x</mi></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi>R</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munder><mi>lim</mi><mrow><mi>y</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>f</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>-</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow><mi>y</mi></mfrac><mo>=</mo><munder><mi>lim</mi><mrow><mi>y</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mi>k</mi><mi>x</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>k</mi><mi>x</mi></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi>R</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>+</mo><mi>C</mi></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi>R</mi></math>  [by integration]</p><p>But <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>2</mn></mfenced><mo>=</mo><mn>8</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mn>2</mn><mo>=</mo><mfrac><mi>k</mi><mn>2</mn></mfrac><mo>+</mo><mi>C</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>=</mo><mn>2</mn><mi>k</mi><mo>+</mo><mi>C</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mn>0</mn></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi>R</mi></math></p><p>So, option (a) is correct.</p></div>


tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
title:
let-f-r-r-be-a-function-such-that-f-x-y-f-x-f-y-x-y-r-if-f-x-is-differentiable-at-x-0-then-which-one-of-the-following-is-incorrect

slug:
let-f-r-r-be-a-function-such-that-f-x-y-f-x-f-y-x-y-r-if-f-x-is-differentiable-at-x-0-then-which-one-of-the-following-is-incorrect-77894

question:
<div class="question">Let $f: R \rightarrow R$ be a function such that $f(x+y)=f(x)+f(y), \forall x, y \in \mathbb{R}$
<br/> <br/>If $\mathrm{f}(\mathrm{x})$ is differentiable at $\mathrm{x}=0$, then which one of the following is incorrect?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{f}(\mathrm{x})$ is continuous, $\forall \mathrm{x} \in \mathrm{R}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{f}^{\prime}(\mathrm{x})$ is constant, $\forall \mathrm{x} \in \mathrm{R}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{f}(\mathrm{x})$ is differentiable, $\forall \mathrm{x} \in \mathrm{R}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{f}(\mathrm{x})$ is differentiable only in a finite interval containing zera.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{f}(\mathrm{x})$ is differentiable only in a finite interval containing zera.</span> </div>

answer:
<div class="solution">Let $f(x+y)=f(x)+f(y), \forall x, y \in R$
<br/> <br/>Put $x=0=y$
<br/> <br/>$\Rightarrow f(0)=f(0)+f(0)$
<br/> <br/>$\Rightarrow f(0)=0$
<br/> <br/>Now, $f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}$
<br/> <br/>$f^{\prime \prime}(0)=\lim _{h \rightarrow 0} \frac{f(h)}{h}$
<br/> <br/>Now, $f(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{f(x)+f(h)-f(x)}{h}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(\mathrm{h})}{\mathrm{h}}=\mathrm{f}^{\prime}(0) \\
<br/> <br/>\Rightarrow \mathrm{f}(\mathrm{x})=\mathrm{x} \mathrm{f}(0)+\mathrm{C}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>But $f(0)=0$
<br/> <br/>$$
<br/> <br/>\therefore \mathrm{C}=0
<br/> <br/>$$
<br/> <br/>Hence, $f(x)=x f(0), \forall x \in R$
<br/> <br/>Clearly, $\mathrm{f}(\mathrm{x})$ is everywhere continuous and differentiable and $\mathrm{f}^{\prime}(\mathrm{x})$ is constant.
<br/> <br/>$\forall \mathrm{x} \in \mathrm{R}$</div>


title: let-f-x-1-x-2-1-1-x-2-x-2-d-x-and-f-0-0-then-the-value-of-f-1-be
slug: let-f-x-1-x-2-1-1-x-2-x-2-d-x-and-f-0-0-then-the-value-of-f-1-be-38127
tags: ['Mathematics', 'Indefinite Integration', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=\int \frac{x^2 d x}{\left(1+x^2\right)\left(1+\sqrt{\left.1+x^2\right)}\right.}$ and $f(0)=0$, 
<br/>then the value of $f(1)$ be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$\log (1+\sqrt{2})$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$\log (1+\sqrt{2})-\frac{\pi}{4}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$\log (1+\sqrt{2})+\frac{\pi}{2}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$\log (1+\sqrt{2})-\frac{\pi}{4}$
<br/></span> </div>

answer:
<div class="solution">$f(x)=\int \frac{x^2 d x}{\left(1+x^2\right)\left(1+\sqrt{1+x^2}\right)}$
<br/>Let $x=\tan \theta$
<br/>$\Rightarrow \quad d x=\sec ^2 \theta d \theta=\left(1+x^2\right) d \theta$
<br/>$f(x)=\int \frac{x^2 d x}{\left(1+x^2\right)\left(1+\sqrt{1+x^2}\right)}$
<br/>$=\int \frac{\tan ^2 \theta \sec ^2 \theta d \theta}{\sec ^2 \theta(1+\sec \theta)}=\int \frac{\tan ^2 \theta d \theta}{1+\sec \theta}$
<br/>$=\int \frac{\sin ^2 \theta d \theta}{\cos \theta(1+\cos \theta)}$
<br/>$=\int \frac{1-\cos ^2 \theta d \theta}{\cos \theta(1+\cos \theta)}$
<br/>$=\int \frac{(1-\cos \theta) d \theta}{\cos \theta}$
<br/>$=\int \sec \theta d \theta-\int d \theta$
<br/>$=\log \left(x+\sqrt{1+x^2}\right)-\tan ^{-1} x+C$
<br/>$\therefore \quad f(0)=\log (0+\sqrt{1+0})-\tan ^{-1}(0)+C$
<br/>$0=\log 1-0+C$
<br/>$\Rightarrow \quad C=0$
<br/>$\therefore \quad f(1)=\log \left(1+\sqrt{1+1^2}\right)-\tan ^{-1}(1)$
<br/>$=\log (1+\sqrt{2})-\frac{\pi}{4}$</div>


title: let-f-x-2-x-3-21-x-2-60-x-41-then
slug: let-f-x-2-x-3-21-x-2-60-x-41-then-52764
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>21</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>60</mn><mi>x</mi><mo>+</mo><mn>41</mn></math>, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is decreasing in <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is decreasing in <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mo>∞</mo><mo>,</mo><mn>2</mn></mrow></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is increasing in <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mo>∞</mo><mo>,</mo><mn>1</mn></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is increasing in <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mo>∞</mo><mo>,</mo><mn>2</mn></mrow></mfenced></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is decreasing in <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mo>∞</mo><mo>,</mo><mn>2</mn></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>21</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>60</mn><mi>x</mi><mo>+</mo><mn>41</mn></math></p><p>On differentiating w.r.t. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>42</mn><mi>x</mi><mo>-</mo><mn>60</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>6</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>6</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>&lt;</mo><mn>2</mn><mo> </mo><mi>or</mi><mo> </mo><mi>x</mi><mo>&gt;</mo><mn>5</mn><mo>,</mo><mo> </mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>&lt;</mo><mn>0</mn></math> i.e., <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is decreasing</p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mn>5</mn><mo>,</mo><mo> </mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>&gt;</mo><mn>0</mn></math> i.e., <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is increasing</p></div>


title: let-f-x-3-c-o-s-2-x-1-s-i-n-3-x-x-2-p-x-2-2-x-2-q-1-s-i-n-x-x-2-if-f-x-is-continuous-at-x-2-p-q
tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
slug:
let-f-x-3-c-o-s-2-x-1-s-i-n-3-x-x-2-p-x-2-2-x-2-q-1-s-i-n-x-x-2-if-f-x-is-continuous-at-x-2-p-q-68035

question:
<div class="question">Let $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}\frac{1-\sin ^{3} \mathrm{x}}{3 \cos ^{2} \mathrm{x}}, \mathrm{x}&lt;\frac{\pi}{2} \\ \mathrm{p}, \quad \mathrm{x}=\frac{\pi}{2} \\ \frac{\mathrm{q}(1-\sin \mathrm{x})}{(\pi-2 \mathrm{x})^{2}}, \mathrm{x}&gt;\frac{\pi}{2}\end{array}\right.$
<br/> <br/>If $f(x)$ is continuous at $x=\frac{\pi}{2},(p, q)=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">(1,4)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\frac{1}{2}, 2\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(\frac{1}{2}, 4\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{1}{2}, 4\right)$</span> </div>

answer:
<div class="solution">$\mathrm{f}\left[(\pi / 2)^{-}\right]=\lim _{\mathrm{h} \rightarrow 0} \frac{1-\sin ^{3}[(\pi / 2)-\mathrm{h}]}{3 \cos ^{2}[(\pi / 2)-\mathrm{h}]}$
<br/> <br/>$\quad=\lim _{\mathrm{h} \rightarrow 0} \frac{1-\cos ^{3} \mathrm{~h}}{3 \sin ^{2} \mathrm{~h}}=\frac{1}{2}$
<br/> <br/>$\mathrm{f}\left[(\pi / 2)^{+}\right]=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{q}[1-\sin \{(\pi / 2)+\mathrm{h}\}]}{[\pi-2\{(\pi / 2)+\mathrm{h}\}]^{2}}$
<br/> <br/>$=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{q}(1-\cosh )}{4 \mathrm{~h}^{2}}=\frac{\mathrm{q}}{8}$
<br/> <br/>$\therefore \mathrm{p}=\frac{1}{2}=\frac{\mathrm{q}}{8} \Rightarrow \mathrm{p}=\frac{1}{2}, \mathrm{q}=4$</div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$x+C$</span> </div>
source_db: MarksBatch2_P1.db
title:
let-f-x-be-a-polynomial-of-degree-three-satisfying-f-0-1-and-f-1-0-also-0-is-a-stationary-point-of-f-x-if-f-x-does-not-have-an-extremum-at-x-0-then-th

slug:
let-f-x-be-a-polynomial-of-degree-three-satisfying-f-0-1-and-f-1-0-also-0-is-a-stationary-point-of-f-x-if-f-x-does-not-have-an-extremum-at-x-0-then-th-52729

question:
<div class="question">Let $f(x)$ be a polynomial of degree three satisfying $\mathrm{f}(0)=-1$ and $\mathrm{f}(1)=0 . \mathrm{Also}, 0$ is a stationary point of $f(x)$ If $f(x)$ does not have an extremum at $x=0$, then the value of $\int \frac{f(x)}{x^{3}-1} d x$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{x^{2}}{2}+C$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x+C$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{x^{3}}{6}+C$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">Let $f(x)=a x^{3}+b x^{2}+c x+d$
<br/> <br/>Put $x=0$ and $x=1$
<br/> <br/>Then, we get $f(0)=-1$ and $f(1)=0$ $\Rightarrow \mathrm{d}=-1$ and $\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}=0$
<br/> <br/>$\Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}=1$
<br/> <br/>It is given that $x=0$ is a stationary point of $\mathrm{f}(\mathrm{x})$, but it is not a point of extremum.
<br/> <br/>Therefore, $f(0)=0=f^{\prime \prime}(0)$ and $f^{\prime \prime \prime}(0)=0$
<br/> <br/>Now, $\mathrm{f}(\mathrm{x})=\mathrm{ax}^{3}+\mathrm{bx}^{2}+\mathrm{cx}+\mathrm{d}$
<br/> <br/>$\Rightarrow f(x)=3 a x^{2}+2 b x+c$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>f^{\prime \prime}(x)=6 a x+2 b \text { and } f^{\prime \prime}(x)=6 a \\
<br/> <br/>f^{\prime}=0, f^{\prime \prime}(0)=0 \text { and } f^{\prime \prime \prime}(0)=0 \neq 0
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>$\Rightarrow c=0, b=0$ and $a \neq 0$
<br/> <br/>From Eqs. (i) and (ii), we get $\mathrm{a}=1, \mathrm{~b}=\mathrm{c}=0$ and $\mathrm{d}=-1$
<br/> <br/>Put these values in $\mathrm{f}(\mathrm{x})$ we get $f(x)=x^{3}-1$
<br/> <br/>Hence, $\int \frac{\mathrm{f}(\mathrm{x})}{\mathrm{x}^{3}-1} \mathrm{dx}=\int \frac{\mathrm{x}^{3}-1}{\mathrm{x}^{3}-1} \mathrm{~d} \mathrm{x}=\int 1 \mathrm{~d} \mathrm{x}=\mathrm{x}+\mathrm{C}$</div>


title: let-f-x-c-x-d-a-x-b-then-fof-x-x-provided-that
slug: let-f-x-c-x-d-a-x-b-then-fof-x-x-provided-that-58866
question: <div class="question">Let $f(x)=\frac{a x+b}{c x+d},$ then fof $(x)=x,$ provided that:</div>
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{d}=-\mathrm{a}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{d}=\mathrm{a}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad a=b=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad a=b=c=d=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{d}=-\mathrm{a}$</span> </div>

answer:
<div class="solution">$f(x)=\frac{a x+b}{c x+d}$
<br/> <br/>fof $(x)=\frac{a\left\{\frac{a x+b}{c x+d}\right\}+b}{c\left\{\frac{a x+b}{c x+d}\right\}+d} \Rightarrow \frac{a^{2} x+a b+b c x+b d}{a c x+b c+c d x+d^{2}}=x$
<br/> <br/>$\Rightarrow(a c+d c) x^{2}+\left(b c+d^{2}-b c-a^{2}\right) x$
<br/> <br/>$-a b-b d=0, \quad \forall x \in R$
<br/> <br/>$\Rightarrow(a+d) c=0, d^{2}-a^{2}=0$ and $(a+d) b=0$
<br/> <br/>$\Rightarrow a+d=0$</div>


title: let-f-x-x-1-sin-x-1-1-if-x-1-0-if-x-1-then-which-one-of-the-following-is-true
slug: let-f-x-x-1-sin-x-1-1-if-x-1-0-if-x-1-then-which-one-of-the-following-is-true-60554
tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f(x)=\left\{\begin{array}{c}(x-1) \sin \frac{1}{x-1} \text { if } x \neq 1 \\ 0 \quad \text { if } x=1\end{array}\right.$
<br/> <br/>Then which one of the following is true?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f$ is differentiable at $x=0$ and $x=1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f$ is differentiable at $x=0$ but not at $x=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f$ is differentiable at $x=1$ but not at $x=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f$ is neither differentiable at $x=0$ nor at $x=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f$ is differentiable at $x=0$ but not at $x=1$</span> </div>

answer:
<div class="solution">We have; $f(x)=\left\{\begin{array}{ll}(x-1) \sin \left(\frac{1}{x-1}\right) &amp; \text { if } x \neq 1 \\ 0 &amp; \text { if } x=1\end{array}\right.$
<br/> <br/>$R f^{\prime}(1)=\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}$
<br/> <br/>$=\lim _{h \rightarrow 0} \frac{h \sin \frac{1}{h}-0}{h}=\lim _{h \rightarrow 0} \sin \frac{1}{h}$
<br/> <br/>which does not exist.
<br/> <br/>$\therefore f$ is not differentiable at $x=1$ Also $\left.f^{\prime}(0)=\sin \frac{1}{(x-1)}-\frac{x-1}{(x-1)^{2}} \cos \left(\frac{1}{x-1}\right)\right]_{x=0}$
<br/> <br/>$=-\sin 1+\cos 1$
<br/> <br/>$\therefore f$ is differentiable at $x=0$</div>


title: let-f-x-x-5-1-x-3-1-g-x-x-2-1-x-2-x-1-and-let-h-x-be-such-that-f-x-g-x-h-x-then-limh-x-1-h-x-is
tags: ['Mathematics', 'Limits', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P1.db
slug:
let-f-x-x-5-1-x-3-1-g-x-x-2-1-x-2-x-1-and-let-h-x-be-such-that-f-x-g-x-h-x-then-limh-x-1-h-x-is-60538

question:
<div class="question">$\quad \operatorname{Let} f(x)=\left(x^{5}-1\right)\left(x^{3}+1\right), g(x)=\left(x^{2}-1\right)\left(x^{2}-x+1\right)$
<br/> <br/>and let $\mathrm{h}(\mathrm{x})$ be such that $\mathrm{f}(\mathrm{x})=\mathrm{g}(\mathrm{x}) \mathrm{h}(\mathrm{x})$. Then
<br/> <br/>$\operatorname{limh}_{x \rightarrow 1} \mathrm{~h}(\mathrm{x})$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given $f(x)=g(x) h(x)$
<br/> <br/>$\Rightarrow h(x)=\frac{f(x)}{g(x)}$
<br/> <br/>$\Rightarrow \lim _{x \rightarrow 1} h(x)=\lim _{x \rightarrow 1} \frac{f(x)}{g(x)}$
<br/> <br/>$\Rightarrow \lim _{x \rightarrow 1} \frac{\left(x^{5}-1\right)\left(x^{3}+1\right)}{\left(x^{2}-1\right)\left(x^{2}-x+1\right)}$
<br/> <br/>$=\lim _{x \rightarrow 1} \frac{x^{5}-1^{5}}{x-1}=5 \times 1^{4}=5$</div>


title: let-f-x-x-p-sin-1-x-x-0-0-x-0-then-f-x-is-continuous-but-not-differentiable-at-x-0-if
slug: let-f-x-x-p-sin-1-x-x-0-0-x-0-then-f-x-is-continuous-but-not-differentiable-at-x-0-if-25647
tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced close="" open="{"><mtable columnalign="left"><mtr><mtd><mtable><mtr><mtd><msup><mi>x</mi><mi>p</mi></msup><mi>sin</mi><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>,</mo></mtd><mtd><mi>x</mi><mo>≠</mo><mn>0</mn></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo></mtd><mtd><mi>x</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mtd></mtr></mtable></mfenced></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is continuous but not differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>, if</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&lt;</mo><mn>0</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>p</mi><mo>≤</mo><mn>1</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>≥</mo><mn>1</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>p</mi><mo>≤</mo><mn>1</mn></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced close="" open="{"><mtable columnalign="left"><mtr><mtd><mtable><mtr><mtd><msup><mi>x</mi><mi>p</mi></msup><mi>sin</mi><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>,</mo></mtd><mtd><mi>x</mi><mo>≠</mo><mn>0</mn></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo></mtd><mtd><mi>x</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mtd></mtr></mtable></mfenced></math></p><p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> to be continuous at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mn>0</mn></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mrow><msup><mi>x</mi><mi>p</mi></msup><mi>sin</mi><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><msup><mi>x</mi><mi>p</mi></msup></mfenced><mo>×</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mrow><mi>sin</mi><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><msup><mi>x</mi><mi>p</mi></msup></mfenced><mo>×</mo><mfenced><mrow><mtext>value between </mtext><mo>-</mo><mn>1</mn><mtext> to</mtext><mo> </mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi>p</mi><mo>&gt;</mo><mn>0</mn></math></p><p>Now,</p><p>RHD at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>RHD at </mtext><mfenced><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced close="]" open="["><mfrac><mrow><msup><mi>h</mi><mi>p</mi></msup><mi>sin</mi><mstyle displaystyle="true"><mfrac><mn>1</mn><mi>h</mi></mfrac></mstyle><mo>-</mo><mn>0</mn></mrow><mi>h</mi></mfrac></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced close="]" open="["><mrow><msup><mi>h</mi><mrow><mi>p</mi><mo>-</mo><mn>1</mn></mrow></msup><mi>sin</mi><mfrac><mn>1</mn><mi>h</mi></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>LHD at </mtext><mfenced><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced close="]" open="["><mfrac><mrow><msup><mfenced><mrow><mo>-</mo><mi>h</mi></mrow></mfenced><mi>p</mi></msup><mi>sin</mi><mfenced><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mi>h</mi></mfrac></mstyle></mrow></mfenced><mo>-</mo><mn>0</mn></mrow><mrow><mo>-</mo><mi>h</mi></mrow></mfrac></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced close="]" open="["><mrow><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mi>p</mi></msup><msup><mfenced><mrow><mo>-</mo><mi>h</mi></mrow></mfenced><mrow><mi>p</mi><mo>-</mo><mn>1</mn></mrow></msup><mi>sin</mi><mfenced><mfrac><mn>1</mn><mi>h</mi></mfrac></mfenced></mrow></mfenced></math></p><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is not differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>, therefore</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>≤</mo><mn>1</mn></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>p</mi><mo>≤</mo><mn>1</mn></math>.</p></div>


title: let-f-x-x-x-x-z-denotes-greatest-integer-function-then-f-x-is-equal-to
slug: let-f-x-x-x-x-z-denotes-greatest-integer-function-then-f-x-is-equal-to-29323
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>x</mi><mfenced close="]" open="["><mi>x</mi></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>∉</mo><mi>Z</mi></math>, (<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mo>·</mo><mo>]</mo></math> denotes greatest integer function), then <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mi>x</mi></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced close="]" open="["><mi>x</mi></mfenced></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mi>x</mi></mfenced></math></span> </div>

answer:
<div class="solution"><p>We have, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>x</mi><mfenced close="]" open="["><mi>x</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfenced close="]" open="["><mi>x</mi></mfenced><mo> </mo></math>is an integer. </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>p</mi><mi>x</mi><mo>  </mo><mfenced close="]" open="["><mrow><mo>∵</mo><mo>[</mo><mi>x</mi><mo>]</mo><mo>=</mo><mi>p</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>p</mi><mo>=</mo><mfenced close="]" open="["><mi>x</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced close="]" open="["><mi>x</mi></mfenced></math></p></div>


title: let-f-x-y-f-x-f-y-for-all-x-y-where-f-0-0-if-f-5-2-and-f-0-3-then-f-5-is-equal-to
slug: let-f-x-y-f-x-f-y-for-all-x-y-where-f-0-0-if-f-5-2-and-f-0-3-then-f-5-is-equal-to-67421
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">Let $f(x+y)=f(x) \cdot f(y)$ for all $x, y$ where $f(0) \neq 0$ If $f(5)=2$ and $f^{\prime}(0)=3,$ then $f^{\prime}(5)$ is equal to-</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">6</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">$f^{\prime}(5)=\lim _{h \rightarrow 0} \frac{f(5+h)-f(5)}{h}$
<br/>$=\lim _{h \rightarrow 0} \frac{f(5+h)-f(5+0)}{h}$
<br/>$=\lim _{h \rightarrow 0} \frac{f(5) \cdot f(h)-f(5)+f(0)}{h}$
<br/>$(\because f(x+y)=f(x), f(y)$ for all $x, y)$
<br/>$=\left(\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}\right) \cdot f(5)=f^{\prime}(0) \cdot f(5)$
<br/>$=3 \times 2=6$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
title:
let-fxfracleftex1right2sin-leftfracxaright-log-left1fracx4right-for-x-neq-0-and-f012-if-mathrmf-is-continuous-at-x0-then-the-value-of-a-is-equal-to

slug:
let-fxfracleftex1right2sin-leftfracxaright-log-left1fracx4right-for-x-neq-0-and-f012-if-mathrmf-is-continuous-at-x0-then-the-value-of-a-is-equal-to-90106

question:
<div class="question">Let \(f(x)=\frac{\left(e^x-1\right)^2}{\sin \left(\frac{x}{a}\right) \log \left(1+\frac{x}{4}\right)}\) for \(x \neq 0\), and \(f(0)=12\). If \(\mathrm{f}\) is continuous at \(x=0\), then the value of \(a\) is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \operatorname{Lt}_{x \rightarrow 0} \frac{\left(e^x-1\right)^2}{\sin \left(\frac{x}{a}\right) \log \left(1+\frac{x}{4}\right)} \\<br/>&amp;=\operatorname{Lt}_{x \rightarrow 0} \frac{\frac{\left(e^x-1\right)^2}{x} \cdot x^2}{\frac{x}{a} \cdot \frac{\sin \left(\frac{x}{a}\right)}{\left(\frac{x}{a}\right)} \cdot \frac{\log \left(1+\frac{x}{4}\right)}{\frac{x}{4}} \cdot \frac{x}{4}} \Rightarrow 4 a=12 \\<br/>&amp; \Rightarrow a=3<br/>\end{aligned}\)</div>


title: let-m-bea-3-3-n-n-singular-matrix-with-det-m-if-m-1-adj-adj-m-ki-then-the-value-of-k-is
slug: let-m-bea-3-3-n-n-singular-matrix-with-det-m-if-m-1-adj-adj-m-ki-then-the-value-of-k-is-72414
tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">Let M bea $3 \times 3 \mathrm{n} \times \mathrm{n}$ -singular matrix with $\operatorname{det}(\mathrm{M})$ $=\alpha$. If $\left[\mathrm{M}^{-1}\right.$ adj $(\mathrm{adj} (\mathrm{M})]=\mathrm{KI}$, then the value of $\mathrm{K}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\alpha$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\alpha^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\alpha^{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\alpha$</span> </div>

answer:
<div class="solution">We know that, $M(\operatorname{adj} M)=|M|$ I
<br/> <br/>Replacing $\mathrm{M}$ by adj $\mathrm{M},$ we get $\operatorname{adj} M[\operatorname{adj}(\operatorname{adj} M)=\operatorname{det}(\operatorname{adj} M) I$
<br/> <br/>$=\operatorname{det}(M) M^{-1}\left[\operatorname{adj}(\operatorname{adj} M)=\alpha^{2} 1\right.$
<br/> <br/>$$
<br/> <br/>\left[\because \mathrm{M}^{-1}=\frac{1}{|\mathrm{M}|} \operatorname{adj}(\mathrm{M})\right]
<br/> <br/>$$
<br/> <br/>$\Rightarrow \alpha \mathrm{M}^{-1}[\operatorname{adj}(\operatorname{adj} M)]=\alpha^{2} \mathrm{I}$
<br/> <br/>$\Rightarrow \mathrm{M}^{-1}[\mathrm{adj}(\operatorname{adj} \mathrm{M})]=\alpha \mathrm{I}$
<br/> <br/>But $M^{-1}[\operatorname{adj}(\operatorname{adj} M)]=K I$
<br/> <br/>Hence, $\mathrm{K}=\alpha$</div>


